Generalized Drazin-meromorphic invertible operators and generalized Kato-meromorphic decomposition

TL;DR

This paper introduces and studies generalized Drazin-meromorphic invertibility and Kato-meromorphic decomposition for bounded linear operators, analyzing their spectra and properties, including conditions for emptiness and spectral differences.

Contribution

It defines new classes of operators and spectra, providing characterizations and relationships, including conditions for polynomial meromorphicity and spectral properties.

Findings

Spectra are empty iff the operator is polynomially meromorphic.

The generalized Kato-meromorphic spectrum differs from the Kato spectrum at most countably.

Characterization of the single-valued extension property for these operators.

Abstract

A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that $T$ admits a generalized Kato-meromorphic decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and the reduction $T_N$ is meromorphic. In this paper we shall investigate such kind of operators and corresponding spectra, the generalized Drazin-meromorphic spectrum and the generalized Kato-meromorphic spectrum, and prove that these spectra are empty if and only if the operator $T$ is polynomially meromorphic. Also we obtain that the generalized Kato-meromorphic spectrum differs from the Kato type spectrum on at most countably many points. Among others, bounded linear operators which can be expressed as a direct sum of a meromorphic operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator are studied. In particular, we shall characterize the single-valued extension property at a point $\lambda_0\in\mathbb{C}$ in the case that $\lambda_0-T$ admits a generalized Kato-meromorphic decomposition. As a consequence we get several results on cluster points of some distinguished parts of the spectrum.

Equations

$$k_n(T)=\dim (N(T)\cap R(T^n))/(N(T)\cap R(T^{n+1}))=\dim (R(T)+N(T^{n+1}))/(R(T)+N(T^n)).$$

$$\mathbf{B}{\mathbf{R}}_1(X)=\mathbf{B}{\mathcal{B}}_{+}(X)=LD(X),\mathbf{B}{\mathbf{R}}_2(X)=\mathbf{B}{\mathcal{B}}_{-}(X)=RD(X),$$

$$\mathbf{B}{\mathbf{R}}_3(X)=\mathbf{B}\mathcal{B}(X)=D(X)$$

$$\mathbf{B}{\Phi }_{+}(X)=L{D}^e(X),\mathbf{B}{\Phi }_{-}(X)=R{D}^e(X).$$

$${\sigma }_{B{\Phi }_{+}}(T)$$

$${\sigma }_{B{\Phi }_{-}}(T)$$

$${\sigma }_{B\Phi }(T)$$

$${\sigma }_{B{\mathcal{W}}_{+}}(T)$$

$${\sigma }_{B{\mathcal{W}}_{-}}(T)$$

$${\sigma }_{B\mathcal{W}}(T)$$

$${\sigma }_{B{\mathcal{B}}_{+}}(T)$$

$${\sigma }_{B{\mathcal{B}}_{-}}(T)$$

$${\sigma }_{B\mathcal{B}}(T)$$

$$\Vert y{\Vert }_d=\inf \{\Vert x\Vert :x\in X,y=T^{d}x\}.$$

$${\sigma }_{TUD}(T)=\{\lambda \in \mathbb{C}:T-\lambda \text{ does not have TUD}\}.$$

$${\sigma }_{Kt}(T)$$

$${\sigma }_{gK}(T)$$

$${\sigma }_{gKR}(T)$$

$$\gamma (T)=\underset{x\notin N(T)}{\inf }\frac{\Vert Tx\Vert }{dist(x,N(T))}.$$

$$TS=ST,STS=S,TST-T\mathrm{is}\mathrm{nilpotent}.$$

$$TS=ST,STS=S,TST-T\mathrm{is}\mathrm{quasinilpotent}.$$

$$TS=ST,STS=S,TST-T\text{is meromorphic}.$$

$$\sigma (T)\mathrm{does}\mathrm{not}\mathrm{cluster}\mathrm{at}{\lambda }_0⟹T\mathrm{and}{T}^{\prime }\mathrm{have}\mathrm{the}\mathrm{SVEP}\mathrm{at}{\lambda }_0,$$

$${\sigma }_{ap}(T)\mathrm{does}\mathrm{not}\mathrm{cluster}\mathrm{at}{\lambda }_0⟹T\mathrm{has}\mathrm{the}\mathrm{SVEP}\mathrm{at}{\lambda }_0,$$

$${\sigma }_{su}(T)\mathrm{does}\mathrm{not}\mathrm{cluster}\mathrm{at}{\lambda }_0⟹T^{\prime }\mathrm{has}\mathrm{the}\mathrm{SVEP}\mathrm{at}{\lambda }_0.$$

$${\sigma }_{\mathbf{gD}\mathcal{M}}(T)$$

$${\sigma }_{\mathbf{gK}\mathcal{M}}(T)$$

$$\mathrm{ind}(T)$$

$$T\mathrm{is}\mathrm{Kato}\text{mbox}and0\notin \mathrm{int}{\sigma }_{\mathbf{B}{\mathbf{R}}_{\mathbf{i}}}(T)⟹T\in {\mathbf{R}}_{\mathbf{i}}.$$

$$0<|\lambda |<ϵ⟹c_n^{\prime }(T-\lambda I)=c_0^{\prime }(T)=\alpha (T)\mathrm{for}\mathrm{all}n\ge 0.$$

$$0<|\lambda |<ϵ⟹$$

$$c_n(T-\lambda I)=c_0(T)=\beta (T)\mathrm{and}{c}_n^{\prime }(T-\lambda I)=c_0^{\prime }(T)=\alpha (T)\mathrm{for}\mathrm{all}n\ge 0.$$

$$c_n^{\prime }(T-{\lambda }_{0}I)=\dim (N(T-{\lambda }_{0}I)\cap R((T-{\lambda }_{0}I{)}^n))<\infty$$

$$\mathrm{ind}(T)$$

$$N((T_{N^{\perp }}^{\prime }{)}^n)\subset R(T_{N^{\perp }}^{\prime }),\mathrm{for}\mathrm{every}n\in \mathbb{N}.$$

$$R((T^{\prime }{P}_M^{\prime }{)}^n)=R(({T^{\prime }}_{N^{\perp }}{)}^n).$$

$$N((T_{N^{\perp }}^{\prime }{)}^n)$$

$$(R(T_M^n)+N{)}^{\perp }\subset R((T{P}_M{)}^{\prime })=N(T{P}_M{)}^{\perp },\mathrm{for}\mathrm{every}n\in \mathbb{N},$$

Full text

2010 Mathematics subject classification: 47A53, 47A10.

Key words and phrases: Banach space; Kato operators; meromorphic operators; polynomially meromorphic operators; single valued extension property; semi-Fredholm spectra; semi-B-Fredholm spectra.

1 Introduction and preliminaries

Throughout this paper $\mathbb{N}\,(\mathbb{N}_{0})$ denotes the set of all positive (non-negative) integers, $\mathbb{C}$ denotes the set of all complex numbers and $L(X)$ denotes the Banach algebra of bounded linear operators acting on an infinite dimensional complex Banach space $X$. The group of all invertible operators is denoted by $L(X)^{-1}$, while set of all bounded below (surjective) operators is denoted by $\mathcal{J}(X)$ ($\mathcal{S}(X)$). For $T\in L(X)$ we shall denote by $\sigma(T)$, $\sigma_{ap}(T)$ and $\sigma_{su}(T)$ its spectrum, approximate point spectrum and surjective spectrum, respectively. Also, we shall write $\alpha(T)$ for the dimension of the kernel $N(T)$ and $\beta(T)$ for the codimension of the range $R(T)$. We call $T\in L(X)$ an upper semi-Fredholm operator if $\alpha(T)<\infty\text{ and }R(T)\text{ is closed}$, but if $\beta(T)<\infty$, then $T$ is a lower semi-Fredholm operator. We use $\Phi_{+}(X)$ (resp. $\Phi_{-}(X)$) to denote the set of upper (resp. lower) semi-Fredholm operators. An operator $T\in L(X)$ is said to be a semi-Fredholm operator if $T$ is upper or lower semi-Fredholm. If $T$ is semi-Fredholm, the index of $T$, ${\rm ind}(T)$, is defined to be ${\rm ind}(T)=\alpha(T)-\beta(T)$. The set of Fredholm operators is defined as $\Phi(X)=\Phi_{+}(X)\cap\Phi_{-}(X).$ The sets of upper semi-Weyl, lower semi-Weyl and Weyl operators are defined as $\mathcal{W}_{+}(X)=\{T\in\Phi_{+}(X):{\rm ind}(T)\leq 0\}$, $\mathcal{W}_{-}(X)=\{T\in\Phi_{-}(X):{\rm ind}(T)\geq 0\}$ and $\mathcal{W}(X)=\{T\in\Phi(X):{\rm ind}(T)=0\}$, respectively. An operator $T\in L(X)$ is said to be Riesz operator, if $T-\lambda\in\Phi(X)$ for every non-zero $\lambda\in\mathbb{C}$. An operator $T\in L(X)$ is meromorphic if its non-zero spectral points are poles of its resolvent, and in that case we shall write $T\in(\mathcal{M})$. It is well known that $T$ is Riesz operator if and only if every nonzero point of $\sigma(T)$ is a pole of the finite algebraic multiplicity. So, every Riesz operator is meromorphic. We say that $T$ is polinomially meromorphic if there exists non-trivial polynomial $p$ such that $p(T)$ is meromorphic.

For $n\in{\mathbb{N}}_{0}$ we set $c_{n}(T)={\rm dim}R(T^{n})/R(T^{n+1})$ and $c_{n}^{\prime}(T)={\rm dim}N(T^{n+1})/N(T^{n})$. From [11, Lemmas 3.1 and 3.2] it follows that $c_{n}(T)={\rm codim}\,(R(T)+N(T^{n}))$ and $c_{n}^{\prime}(T)={\rm dim}(N(T)\cap R(T^{n}))$. Obviously, the sequences $(c_{n}(T))_{n}$ and $(c_{n}^{\prime}(T))_{n}$ are decreasing. For each $n\in{\mathbb{N}}_{0}$, $T$ induces a linear transformation from the vector space $R(T^{n})/R(T^{n+1})$ to the space $R(T^{n+1})/R(T^{n+2})$: let $k_{n}(T)$ denote the dimension of the null space of the induced map. From [8, Lemma 2.3] it follows that

[TABLE]

From this it is easily seen that $k_{n}(T)=c_{n}^{\prime}(T)-c_{n+1}^{\prime}(T)$ if $c_{n+1}^{\prime}(T)<\infty$ and $k_{n}(T)=c_{n}(T)-c_{n+1}(T)$ if $c_{n+1}(T)<\infty$.

The descent $\delta(T)$ and the ascent $a(T)$ of $T$ are defined by $\delta(T)=\inf\{n\in{\mathbb{N}}_{0}:c_{n}(T)=0\}=\inf\{n\in{\mathbb{N}}_{0}:R(T^{n})=R(T^{n+1})\}$ and $a(T)=\inf\{n\in{\mathbb{N}}_{0}:c^{\prime}_{n}(T)=0\}=\inf\{n\in{\mathbb{N}}_{0}:N(T^{n})=N(T^{n+1})\}$. We set formally $\inf\emptyset=\infty$.

The essential descent $\delta_{e}(T)$ and the essential ascent $a_{e}(T)$ of $T$ are defined by $\delta_{e}(T)=\inf\{n\in{\mathbb{N}}_{0}:c_{n}(T)<\infty\}$ and $a_{e}(T)=\inf\{n\in{\mathbb{N}}_{0}:c^{\prime}_{n}(T)<\infty\}$.

The sets of upper semi-Browder, lower semi-Browder and Browder operators are defined as $\mathcal{B}_{+}(X)=\{T\in\Phi_{+}(X):a(T)<\infty\}$, $\mathcal{B}_{-}(X)=\{T\in\Phi_{-}(X):\delta(T)<\infty\}$ and $\mathcal{B}(X)=\mathcal{B}_{+}(X)\cap\mathcal{B}_{-}(X)$, respectively.

Sets of left and right Drazin invertible operators, respectively, are defined as $LD(X)=\{T\in L(X):a(T)<\infty{\rm\ and\ }R(T^{a(T)+1})\ {\rm is\ closed}\}$ and $RD(X)=\{T\in L(X):\delta(T)<\infty{\rm\ and\ }R(T^{\delta(T)})\ {\rm is\ closed}\}.$ If $a(T)<\infty$ and $\delta(T)<\infty$, then $T$ is called Drazin invertible. By $D(X)$ we denote the set of Drazin invertible operators.

An operator $T\in L(X)$ is a left essentially Drazin invertible operator if $a_{e}(T)<\infty$ and $R(T^{a_{e}(T)+1})$ is closed. If $\delta_{e}(T)<\infty$ and $R(T^{\delta_{e}(T)})$ is closed, then $T$ is called right essentially Drazin invertible. In the sequel $LD^{e}(X)$ (resp. $RD^{e}(X)$) will denote the set of left (resp. right) essentially Drazin invertible operators. For a bounded linear operator T and $n\in{\mathbb{N}}_{0}$ define $T_{n}$ to be the restriction of $T$ to $R(T^{n})$ viewed as a map from $R(T^{n})$ into $R(T^{n})$ (in particular, $T_{0}=T$). If $T\in L(X)$ and if there exists an integer $n$ for which the range space $R(T^{n})$ is closed and $T_{n}$ is Fredholm (resp. upper semi-Fredholm, lower semi-Fredholm, Browder, upper semi-Browder, lower semi-Browder), then $T$ is called a B-Fredholm (resp. upper semi-B-Fredholm, lower semi-B-Fredholm, B-Browder, upper semi-B-Browder, lower semi-B-Browder ) operator. If $T\in L(X)$ is upper or lower semi-B-Fredholm, then T is called semi-B-Fredholm. The index ${\rm ind}(T)$ of a semi-B-Fredholm operator T is defined as the index of the semi-Fredholm operator $T_{n}$. By [4, Proposition 2.1] the definition of the index is independent of the integer n. An operator $T\in L(X)$ is B-Weyl (resp. upper semi-B-Weyl, lower semi-B-Weyl) if $T$ is B-Fredholm and ind(T) = 0 (resp. $T$ is upper semi-B-Fredholm and ${\rm ind}(T)\leq 0$, $T$ is lower semi-B- Fredholm and ${\rm ind}(T)\geq 0$).

We use the following notation:

[TABLE]

We recall that [5, Theorem 3.6]

[TABLE]

For $T\in L(X)$, the upper semi-B-Fredholm spectrum, the lower semi-B-Fredholm spectrum, the B-Fredholm spectrum, the upper semi-B-Weyl spectrum, the lower semi-B-Weyl spectrum, the B-Weyl spectrum, the upper semi-B-Browder spectrum, the lower semi-B-Browder spectrum and the B-Browder spectrum are defined, respectively, by:

[TABLE]

For $T\in L(X)$ and every $d\in{\mathbb{N}}_{0}$, the operator range topology on $R(T^{d})$ is defined by the norm $\|\cdot\|_{d}$ such that for every $y\in R(T^{d})$,

[TABLE]

Operators which have eventual topological uniform descent were introduced by Grabiner in [8]:

Definition 1.1**.**

Let $T\in L(X)$. If there is $d\in{\mathbb{N}}_{0}$ for which $k_{n}(T)=0$ for $n\geq d$, then $T$ is said to have uniform descent for $n\geq d$. If in addition, $R(T^{n})$ is closed in the operator range topology of $R(T^{d})$ for $n\geq d$, then we say that $T$ has eventual topological uniform descent and, more precisely, that $T$ has topological uniform descent (TUD for brevity) for $n\geq d$.

For $T\in L(X)$, the topological uniform descent spectrum is defined by:

[TABLE]

If $M$ is a subspace of $X$ such that $T(M)\subset M$, $T\in L(X)$, it is said that $M$ is $T$-invariant. We define $T_{M}:M\to M$ as $T_{M}x=Tx,\,x\in M$. If $M$ and $N$ are two closed $T$-invariant subspaces of $X$ such that $X=M\oplus N$, we say that $T$ is completely reduced by the pair $(M,N)$ and it is denoted by $(M,N)\in Red(T)$. In this case we write $T=T_{M}\oplus T_{N}$ and say that $T$ is the direct sum of $T_{M}$ and $T_{N}$.

For $T\in L(X)$ we say that it is Kato if $R(T)$ is closed and $N(T)\subset R(T^{n})$ for every $n\in\mathbb{N}$. It is said that $T\in L(X)$ admits a Kato decomposition or $T$ is of Kato type if there exist two closed $T$-invariant subspaces $M$ and $N$ such that $X=M\oplus N$, $T_{M}$ is Kato and $T_{N}$ is nilpotent. If we require that $T_{N}$ is quasinilpotent instead of nilpotent in the definition of the Kato decomposition, then it leads us to the generalized Kato decomposition, abbreviated as GKD [13]. An operator $T\in L(X)$ is said to admit a generalized Kato-Riesz decomposition if there exists a pair $(M,N)\in Red(T)$ such that $T_{M}$ is Kato and $T_{N}$ is Riesz.

For $T\in L(X)$, the Kato type spectrum, the generalized Kato spectrum and the generalized Kato-Riesz spectrum are defined, respectively, by:

[TABLE]

If $K\subset\mathbb{C}$, then $\partial K$ is the boundary of $K$, ${\rm acc}\,K$ is the set of accumulation points of $K$, ${\rm iso}\,K=K\setminus{\rm acc}\,K$ and ${\rm int}\,K$ is the set of interior points of $K$. For $\lambda_{0}\in{\mathbb{C}}$, the open disc, centered at $\lambda_{0}$ with radius $\epsilon$ in ${\mathbb{C}}$, is denoted by $D(\lambda_{0},\epsilon)$.

If $T\in L(X)$ the reduced minimum modulus of a non-zero operators $T$ is defined to be

[TABLE]

If $T=0$, then we take $\gamma(T)=\infty$.

Recall that an operator $T\in L(X)$ is Drazin invertible if there is $S\in L(X)$ such that

[TABLE]

The concept of the generalized Drazin invertible operators was introduced by J. Koliha [12]: an operator $T\in L(X)$ is generalized Drazin invertible in case there is $S\in L(X)$ such that

[TABLE]

Recall that $T$ is generalized Drazin invertible if and only if $0\notin{\rm acc}\,\sigma(T)$, and this is also equivalent to the fact that $T=T_{1}\oplus T_{2}$ where $T_{1}$ is invertible and $T_{2}$ is quasinilpotent. In [20] this concept is further generalized by replacing the third condition in the previous definitions by the condition that $TST-T$ is Riesz, and so it is introduced the concept of generalized Drazin-Riesz invertible operators. It is proved that $T$ is generalized Drazin-Riesz invertible if and only if $T$ admits a generalized Kato-Riesz decomposition and [math] is not an interior point of $\sigma(T)$, and this is also equivalent to the fact that $T=T_{1}\oplus T_{2}$, where $T_{1}$ is invertible and $T_{2}$ is Riesz.

In this paper we further generalize this concept by introducing generalized Drazin-meromorphic invertible operators.

Definition 1.2**.**

An operator $T\in L(X)$ is generalized Drazin-meromorphic invertible, if there exists $S\in L(X)$ such that

[TABLE]

Obviously, every meromorphic operator is generalized Drazin-meromorphic invertible.

Definition 1.3**.**

An operator $T\in L(X)$ is said to admit a generalized Kato-meromorphic decomposition, abbreviated to $GK(\mathcal{M})D$, if there exists a pair $(M,N)\in Red(T)$ such that $T_{M}$ is Kato and $T_{N}$ is meromorphic (i.e. $T_{N}\in(\mathcal{M})$). In that case we shall say that $T$ admits a $GK(\mathcal{M})D(M,N)$.

Definition 1.4**.**

An operator $T\in L(X)$ satisfies $T\in{\bf GD(\mathcal{M})R_{i}}$ if there exist $(M,N)\in Red(T)$ such that $T_{M}\in{\bf R_{i}}$ and $T_{N}\in(\mathcal{M})$, $1\leq i\leq 9$.

In the second section of this paper, using Grabiner’s punctured neighborhood theorem [8, Theorem 4.7], we characterize operators which belong to the set ${\bf GD(\mathcal{M})R_{i}}$, $1\leq i\leq 9$. In particular, we characterize generalized Drazin-meromorphic invertible operators and among other results, we prove that $T$ is generalized Drazin-meromorphic invertible if and only if $T=T_{1}\oplus T_{2}$ where $T_{1}$ is invertible and $T_{2}$ is meromorphic, and this is also equivalent to the fact that $T$ admits a generalized Kato-meromorphic decomposition and [math] is not an interior point of $\sigma(T)$. Also, if $T$ admits a generalized Kato-meromorphic decomposition, then [math] is not an interior point of $\sigma(T)$ if and only if [math] is not an acumulation point of $\sigma_{\bf BB}(T)$.

An operator $T\in L(X)$ is said to have the single-valued extension property at $\lambda_{0}\in{\mathbb{C}}$ (SVEP at $\lambda_{0}$ for breviety) if for every open disc $\mathcal{D}_{\lambda_{0}}$ centerd at $\lambda_{0}$ the only analitic function $f:\mathcal{D}_{\lambda_{0}}\to X$ satisfying $(T-\lambda)f(\lambda)=0$ for all $\lambda\in\mathcal{D}_{\lambda_{0}}$ is the function $f\equiv 0$. An operator $T\in L(X)$ is said to have the SVEP if $T$ has the SVEP at every point $\lambda\in{\mathbb{C}}$. There are implications (see [2], p. 182):

[TABLE]

P. Aiena and E. Rosas gave characterizations of the SVEP at $\lambda_{0}$ in the case that $\lambda_{0}-T$ is of Kato type. Precisely, they proved that if $\lambda_{0}-T$ is of Kato type, then the implications (1.2), (1.3) and (1.4) can be reversed [2]. Q. Jiang and H. Zhong [10] gave further characterizations of the SVEP at $\lambda_{0}$ in the case that $\lambda_{0}-T$ admits a generalized Kato decomposition. They proved that if $\lambda_{0}-T$ admits a GKD, then the following statements are equivalent:

(i) $T$ ($T^{\prime}$) has the SVEP at $\lambda_{0}$;

(ii) $\sigma_{ap}(T)$ ($\sigma_{su}(T)$) does not cluster at $\lambda_{0}$;

(iii) $\lambda_{0}$ is not an interior point of $\sigma_{ap}(T)$ ($\sigma_{su}(T)$),

that is, the implications (1.2), (1.3) and (1.4) can be also reversed in the case that $\lambda_{0}-T$ admits a GKD. In [20] it was showed that if $\lambda_{0}-T$ admits a generalized Kato-Riesz decomposition, then the following statements are equivalent:

(i) $T\ (T^{\prime})\ {\rm has\ the\ SVEP\ at\ }\lambda_{0}$;

(ii) $\sigma_{\mathcal{B}_{+}}(T)\ (\sigma_{\mathcal{B}_{-}}(T))$ does not cluster at $\lambda_{0}$;

(iii) $\lambda_{0}$ is not an interior point of $\sigma_{ap}(T)\ (\sigma_{su}(T))$.

In the second section we give further characterizations of the SVEP at $\lambda_{0}$ always in the case that $\lambda_{0}-T$ admit a generalized Kato-meromorphic decomposition. Precisely, we prove that if $\lambda_{0}-T$ admits a generalized Kato-meromorphic decomposition, then $T$ has the SVEP at $\lambda_{0}$ if and only if $\sigma_{\bf BB_{+}}(T)$ does not cluster at $\lambda_{0}$, and it is precisely when $\lambda_{0}$ is not an interior point of $\sigma_{ap}(T)$. A dual result shows that, always if $\lambda_{0}-T$ admits a generalized Kato-meromorphic decomposition, $T^{\prime}$ has the SVEP at $\lambda_{0}$ if and only if $\sigma_{\bf BB_{-}}(T)$ does not cluster at $\lambda_{0}$, and it is precisely when $\lambda_{0}$ is not an interior point of $\sigma_{su}(T)$.

Also, we prove that if $\lambda_{0}-T$ admits a $GK(\mathcal{M})D$, then $\lambda_{0}$ is not an interior point of $\sigma_{{\bf R}}(T)$ if and only if $\sigma_{{\bf BR}}(T)$ does not cluster at $\lambda_{0}$ where ${\bf R}$ is one of $\Phi_{+},\Phi_{-},\Phi,\mathcal{W}_{+},\mathcal{W}_{-},\mathcal{W}$.

In the third section we investigate corresponding spectra. For $T\in L(X)$, the generalized Drazin-meromorphic spectrum and the generalized Kato-meromorphic spectrum are defined, respectively, by

[TABLE]

It is proved that these spectra are compact and that the generalized Kato-meromorphic spectrum differs from the Kato type spectrum on at most countably many points. We deduce several results on cluster points of some semi-B-Fredholm spectra. Among other results, it is proved that $\partial\,\sigma_{{\bf R}_{i}}(T)\cap{\rm acc}\,\sigma_{{\bf BR}_{i}}(T)\subset\partial\,\sigma_{\bf gK\mathcal{M}}(T)$, $1\leq i\leq 9$. Also we get some results regarding boundaries and connected hulls of corresponding spectra (${\bf GD(\mathcal{M})R_{i}}$ spectra), and obtain that the generalized Drazin-meromorphic spectrum and the generalized Kato-meromorphic spectrum of $T\in L(X)$ are empty in the same time and that this happens if and only if $T$ is polynomially meromorphic.

These results are applied to some concrete operators, amongst them the unilateral weighted right shift operator on $\ell_{p}({\mathbb{N}})$, $1\leq p<\infty$, the forward and backward unilateral shifts on $c_{0}({\mathbb{N}}),c({\mathbb{N}}),\ell_{\infty}({\mathbb{N}})$ or $\ell_{p}({\mathbb{N}})$, $1\leq p<\infty$, arbitrary non-invertible isometry, and $\rm Ces\acute{a}ro$ operator.

2 Generalized Drazin-meromorphic invertible and

generalized Drazin-meromorphic semi-Fredholm

operators

We start with the following auxiliary assertions.

Lemma 2.1**.**

Let $T\in L(X)$. If $T$ is Kato and $T$, $T^{\prime}$ have SVEP at 0, then $T$ is invertible.

Proof.

It follows from [1, Theorem 2.49]. ∎

Lemma 2.2**.**

Let $T\in L(X)$ and let $(M,N)\in Red(T)$. Then $T\in(\mathcal{M})$ if and only if $T_{M}\in(\mathcal{M})$ and $T_{N}\in(\mathcal{M})$.

Proof.

Since $\sigma_{D}(T)=\sigma_{D}(T_{M})\cup\sigma_{D}(T_{N})$, it follows that $\sigma_{D}(T)\subset\{0\}$ if and only if $\sigma_{D}(T_{M})\subset\{0\}$ and $\sigma_{D}(T_{N})\subset\{0\}$. Consequently, $T$ is meromorphic if and only if $T_{M}$ and $T_{N}$ are meromorphic. ∎

Lemma 2.3**.**

Let $T\in L(X)$ and $(M,N)\in Red(T)$. The following statements hold:

(i)* $T\in{\bf BR}_{i}$, $1\leq i\leq 6$, if and only if $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in{\bf BR}_{i}$ and then ${\rm ind}(T)={\rm ind}(T_{M})+{\rm ind}(T_{N})$;*

(ii)* If $7\leq i\leq 9$, $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in{\bf BR}_{i}$, then $T\in{\bf BR}_{i}$.*

(iii)* If $7\leq i\leq 9$, $T\in{\bf BR}_{i}$ and $T_{N}$ is B-Weyl, then $T_{M}\in{\bf BR}_{i}$.*

Proof.

Since $N(T^{n})=N(T_{M}^{n})\oplus N(T_{N}^{n})$ and $R(T^{n})=R(T_{M}^{n})\oplus R(T_{N}^{n})$ for every $n\in{\mathbb{N}}_{0}$, it follows that $c_{n}(T)=c_{n}(T_{M})+c_{n}(T_{N})$ and $c_{n}^{\prime}(T)=c_{n}^{\prime}(T_{M})+c_{n}^{\prime}(T_{N})$. Thus $c_{n}(T)=0$ if and only if $c_{n}(T_{M})=c_{n}(T_{N})=0$, and $c_{n}^{\prime}(T)=0$ if and only if $c_{n}^{\prime}(T_{M})=c_{n}^{\prime}(T_{N})=0$. Also $c_{n}(T)<\infty$ if and only if $c_{n}(T_{M})<\infty$, $c_{n}(T_{N})<\infty$, and $c_{n}^{\prime}(T)<\infty$ if and only if $c_{n}^{\prime}(T_{M})<\infty$, $c_{n}^{\prime}(T_{N})<\infty$. Therefore, $\delta(T)<\infty$ if and only if $\delta(T_{M})<\infty$ and $\delta(T_{N})<\infty$, in which case $\delta(T)=\max\{\delta(T_{M}),\delta(T_{N})\}$, and similarly for the ascent, the essential descent and the essential ascent. As $R(T^{n})$ is closed if and only if $R(T_{M}^{n})$ and $R(T_{N}^{n})$ are closed [10, Lemma 3.3], for every $n\in{\mathbb{N}}_{0}$, we get that $T$ is left Drazin invertible (right Drazin invertible, left (right) essentially Drazin invertible) if and only if $T_{M}$ and $T_{N}$ are left Drazin invertible (right Drazin invertible, left (right) essentially Drazin invertible), which by [5, Theorem 3.6] means that $T\in{\bf BR}_{i}$, $1\leq i\leq 6$, if and only if $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in{\bf BR}_{i}$. In that case, for $n$ large enough, we have that

[TABLE]

(ii) follows from (i).

(iii): Suppose that $T$ is upper semi-B-Weyl and that $T_{N}$ is B-Weyl. Then ${\rm ind}(T_{N})=0$ and from (i) it follows that $T_{M}$ is upper semi-B-Fredholm and ${\rm ind}(T_{M})={\rm ind}(T_{M})+{\rm ind}(T_{N})={\rm ind}(T)\leq 0$. Hence $T_{M}$ is upper semi-B-Weyl.

Similarly for the rest of the cases. ∎

Lemma 2.4**.**

Let $X=X_{1}\oplus X_{2}\dots\oplus X_{n}$ where $X_{1},\ X_{2},\dots,X_{n}$ are closed subspaces of $X$ and let $M_{i}$ be a closed subset of $X_{i}$, $i=1,\dots,n$. Then the set $M_{1}\oplus M_{2}\oplus\dots\oplus M_{n}$ is closed.

Proof.

Consider Banach space $X_{1}\times X_{2}\times\dots\times X_{n}$ provided with the canonical norm $\|(x_{1},\dots,x_{n})\|=\sum_{i=1}^{n}\|x_{i}\|$, $x_{i}\in X_{i}$, $i=1,\dots,n$. Then the map $f:X_{1}\times\dots\times X_{n}\to X_{1}\oplus\dots\oplus X_{n}=X$ defined by $f((x_{1},\dots,x_{n}))=x_{1}+\dots+x_{n}$, $x_{i}\in X_{i}$, $i=1,\dots,n$, is a homeomorphism. Since $M_{1}\times M_{2}\times\dots\times M_{n}$ is closed in $X_{1}\times X_{2}\times\dots\times X_{n}$, it follows that $f(M_{1}\times M_{2}\times\dots\times M_{n})=M_{1}\oplus M_{2}\oplus\dots\oplus M_{n}$ is closed. ∎

Lemma 2.5**.**

Let $T\in L(X)$ and $1\leq i\leq 9$. Then the following implication holds:

[TABLE]

Proof.

We prove the assertion for the cases $i=1$ and $i=6$.

Suppose that $T$ is Kato and that [math] is not an interior point of $\sigma_{\bf BR_{1}}(T)$. Then $T$ has TUD for $n\geq 0$, and from [8, Theorem 4.7] we have that there is an $\epsilon>0$ such that for every $\lambda\in{\mathbb{C}}$ the following implication holds:

[TABLE]

From $0\notin{\rm int}\,\sigma_{\bf BR_{1}}(T)$ it follows that there exists $\lambda_{0}\in{\mathbb{C}}$ such that $0<|\lambda_{0}|<\epsilon$ and $T-\lambda_{0}I$ is a left Drazin invertible operator. This implies that there exists $n\in{\mathbb{N}}_{0}$ such that $c_{n}^{\prime}(T-\lambda_{0}I)=0$. From (2.1) it follows that $\alpha(T)=0$ and since $R(T)$ is closed, we obtain that $T$ is bounded bellow.

Suppose that $T$ is Kato and that [math] is not an interior point of $\sigma_{B\mathcal{W}_{+}}(T)$. According to [8, Theorem 4.7] there is an $\epsilon>0$ such that for every $\lambda\in{\mathbb{C}}$ the following implication holds:

[TABLE]

From $0\notin{\rm int}\,\sigma_{B\mathcal{W}_{+}}(T)$, we have that there exists $\lambda_{0}\in{\mathbb{C}}$ such that $0<|\lambda_{0}|<\epsilon$ and $T-\lambda_{0}I$ is an upper semi-B-Weyl operator. Therefore,

[TABLE]

for $n$ large enough and according to (2.2) we obtain that $\alpha(T)<\infty$. As $R(T)$ is closed, we get that $T$ is upper semi-Fredholm. From

[TABLE]

it follows that $T$ is an upper semi-Weyl operator.

The remaining cases can be proved similarly. ∎

Let $U\subset X$ and $W\subset X^{\prime}$. The annihilator of $U$ is the set $U^{\bot}=\{x^{\prime}\in X^{\prime}:x^{\prime}(u)=0\text{\ for every\ }u\in U\}$, and the annihilator of $W$ is the set ${}^{\bot}W=\{x\in X:w(x)=0\text{\ for every\ }w\in W\}$.

Lemma 2.6**.**

Let $(M,N)\in Red(T)$. Then $T$ admits a $GK(\mathcal{M})D(M,N)$ if and only if $T^{\prime}$ admits a $GK(\mathcal{M})D(N^{\bot},M^{\bot})$.

Proof.

Suppose that $T$ admits a $GK(\mathcal{M})D(M,N)$. Then $T_{M}$ is Kato, $T_{N}\in(\mathcal{M})$ and $(N^{\bot},M^{\bot})\in Red(T^{\prime})$. Let $P_{N}$ be the projection of $X$ onto $N$ along $M$. Then $(M,N)\in Red(TP_{N})$, $TP_{N}=P_{N}T$, $TP_{N}=0\oplus T_{N}$, and so, according to Lemma 2.2, it follows that $TP_{N}\in(\mathcal{M})$.  Consequently, $T^{\prime}P_{N}^{\prime}=P_{N}^{\prime}T^{\prime}\in(\mathcal{M})$, $(N^{\bot},M^{\bot})\in Red(T^{\prime}P_{N}^{\prime})$ and since $R(P_{N}^{\prime})=N(P_{N})^{\bot}=M^{\bot}$, we conclude that $(T^{\prime}P_{N}^{\prime})_{M^{\bot}}={T^{\prime}}_{M^{\bot}}\in(\mathcal{M})$ according to Lemma 2.2. From the proof of Theorem 1.43 in [1] it follows that ${T^{\prime}}_{N^{\bot}}$ is Kato. Therefore, $(N^{\bot},M^{\bot})$ is a $GK(\mathcal{M})D$ for $T^{\prime}$.

Let $T^{\prime}$ admit a $GK(\mathcal{M})D(N^{\bot},M^{\bot})$. Then ${T^{\prime}}_{N^{\bot}}$ is Kato and ${T^{\prime}}_{M^{\bot}}\in(\mathcal{M})$. As $(N^{\bot},M^{\bot})\in Red(T^{\prime}P_{N}^{\prime})$, then $T^{\prime}P_{N}^{\prime}=(T^{\prime}P_{N}^{\prime})_{N^{\bot}}\oplus(T^{\prime}P_{N}^{\prime})_{M^{\bot}}=0\oplus{T^{\prime}}_{M^{\bot}}$, and according to Lemma 2.2 we obtain $T^{\prime}P_{N}^{\prime}\in(\mathcal{M})$, which implies $TP_{N}\in(\mathcal{M})$. Since $TP_{N}=0\oplus T_{N}$, from Lemma 2.2 we get $T_{N}\in(\mathcal{M})$. Let $P_{M}=I-P_{N}$. Since ${T^{\prime}}_{N^{\bot}}$ is Kato, it follows that $R(({T^{\prime}}_{N^{\bot}})^{n})$ is closed ([16, Theorem 12.2]) and

[TABLE]

From $(N^{\bot},M^{\bot})\in Red(T^{\prime}P_{M}^{\prime})$, we have $(T^{\prime}P_{M}^{\prime})^{n}=(T^{\prime}P_{M}^{\prime})_{N^{\bot}}^{n}\oplus(T^{\prime}P_{N}^{\prime})_{M^{\bot}}^{n}={T^{\prime}}_{N^{\bot}}^{n}\oplus 0$ and

[TABLE]

So $R((T^{\prime}P_{M}^{\prime})^{n})$ is closed which implies that $R((TP_{M})^{n})=R(T_{M}^{n})$ is closed. As

[TABLE]

from (2.3) and (2.4) we obtain

[TABLE]

which implies

[TABLE]

From Lemma 2.4 it follows that $R(T_{M}^{n})+N$ is closed and therefore, according to [17, Chapter III, Lemma 3.2], we get ${}^{\bot}{({(R(T_{M}^{n})+N)}^{\bot})}=R(T_{M}^{n})+N$, $n\in{\mathbb{N}}$. Now from (2.5) we obtain

[TABLE]

It implies $N(T_{M})\subset\,R(T_{M}^{n})\ {\rm for\ every\ }n\in{\mathbb{N}}$ and we can conclude that $T_{M}$ is Kato. ∎

Theorem 2.1**.**

The following conditions are equivalent for $T\in L(X)$ and $1\leq i\leq 9$:

(i)* There exists $(M,N)\in Red(T)$ such that $T_{M}\in{\bf R}_{i}$ and $T_{N}\in(\mathcal{M})$, that is $T\in{\bf GD(\mathcal{M})R_{i}}$;*

(ii)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{{\bf R}_{i}}(T)$;*

(iii)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm acc}\,\sigma_{{\bf BR}_{i}}(T)$;*

(iv)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{{\bf BR}_{i}}(T)$.*

Proof.

(i)$\Longrightarrow$(ii), (i)$\Longrightarrow$(iii): Suppose that there exists $(M,N)\in Red(T)$ such that $T_{M}\in{\bf R}_{i}$ and $T_{N}\in(\mathcal{M})$. For $1\leq i\leq 3$, $T_{M}$ is Kato, and so $T$ admits a $GK(\mathcal{M})D$. For $4\leq i\leq 9$, from [16, Theorem 16.20] there exists $(M_{1},M_{2})\in Red(T_{M})$ such that ${\rm dim}M_{2}<\infty$, $T_{M_{1}}$ is Kato and $T_{M_{2}}$ is nilpotent. Then for $N_{1}=M_{2}\oplus N$ we have that $N_{1}$ is a closed subspace and $T_{N_{1}}=T_{M_{2}}\oplus T_{N}\in(\mathcal{M})$ by Lemma 2.2. So $T$ admits a $GK(\mathcal{M})D$.

From $T_{M}\in{\bf R}_{i}$ it follows that $0\notin\sigma_{{\bf R}_{i}}(T_{M})$ and there exists an $\epsilon>0$ such that $D(0,\epsilon)\cap\sigma_{{\bf R}_{i}}(T_{M})=\emptyset$. As $T_{N}\in(\mathcal{M})$, we have that $\sigma_{{\bf R}_{i}}(T_{N})$ is at most countable (with [math] as its only possible limit point) and since $\sigma_{{\bf R}_{i}}(T)\subset\sigma_{{\bf R}_{i}}(T_{M})\cup\sigma_{{\bf R}_{i}}(T_{N})$, we conclude that $0\notin{\rm int}\,\sigma_{{\bf R}_{i}}(T)$.

From $T_{M}\in{\bf R}_{i}$ it follows that there exists $\epsilon>0$ such that for every $\lambda\in{\mathbb{C}}$ satisfying $|\lambda|<\epsilon$ we have that $T_{M}-\lambda I_{M}\in{\bf R_{i}}\subset{\bf BR_{i}}$. Since $T_{N}\in(\mathcal{M})$, we have that $T_{N}-\lambda I_{N}$ is Drazin invertible (and hence belongs to ${\bf BR_{i}}$) for every $\lambda\in{\mathbb{C}}$ such that $0<|\lambda|<\epsilon$. According to Lemma 2.3 (i),(ii), we obtain that $T-\lambda I\in{\bf BR_{i}}$ for every $\lambda\in{\mathbb{C}}$ such that $0<|\lambda|<\epsilon$, and so $0\notin{\rm acc}\,\sigma_{{\bf BR}_{i}}(T)$.

(ii)$\Longrightarrow$(iv): It follows from the inclusion $\sigma_{{\bf BR_{i}}}(T)\subset\sigma_{{\bf R_{i}}}(T)$.

(iii)$\Longrightarrow$(iv): It is obvious.

(iv)$\Longrightarrow$(i): Suppose that $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{{\bf BR}_{i}}(T)$. Then there exists a decomposition $(M,N)\in Red(T)$ such that $T_{M}$ is Kato and $T_{N}\in(\mathcal{M})$. From $0\notin{\rm int}\,\sigma_{{\bf BR}_{i}}(T)$ it follows that for every $\epsilon>0$ there exists $\lambda\in{\mathbb{C}}$ such that $0<|\lambda|<\epsilon$ and $T-\lambda I\in{\bf BR_{i}}$. Since $T_{N}\in(\mathcal{M})$ we have that $T_{N}-\lambda I_{N}$ is Drazin invertible (and hence B-Weyl), and therefore from Lemma 2.3 we get that $T_{M}-\lambda I_{M}\in{\bf BR_{i}}$. Thus $0\notin{\rm int}\,\sigma_{{\bf BR}_{i}}(T_{M})$. As $T_{M}$ is Kato, from Lemma 2.5 it follows that $T_{M}\in{\bf R_{i}}$. ∎

Theorem 2.2**.**

The following conditions are equivalent for $T\in L(X)$ and $1\leq i\leq 9$:

(i)* There exists $(M,N)\in Red(T)$ such that $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in(\mathcal{M})$ ;*

(ii)* There exists a bounded projection $P$ on $X$ which commutes with $T$ such that $T+P\in{\bf BR}_{i}$ and $TP\in(\mathcal{M})$;*

(iii)* There exists a bounded projection $P$ on $X$ which commutes with $T$ such that $T(I-P)+P\in{\bf BR}_{i}$ and $TP\in(\mathcal{M})$.*

Proof.

(i)$\Longrightarrow$(ii): Suppose that there exists $(M,N)\in Red(T)$ such that $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in(\mathcal{M})$. Then the projection $P\in L(X)$ such that $N(P)=M$ and $R(P)=N$ satisfies $TP=PT$. Since $T_{N}\in(\mathcal{M})$, $TP=0\oplus T_{N}\in(\mathcal{M})$ by Lemma 2.2. Again, since $T_{N}\in(\mathcal{M})$, it follows that $T_{N}+I_{N}$ is Drazin invertible, i.e. B-Browder, and so $T+P=T_{M}\oplus(T_{N}+I_{N})\in{\bf BR_{i}}$ according to Lemma 2.3.

(ii)$\Longrightarrow$(iii): If (ii) holds, then for $M=(I-P)X$ and $N=PX$ from $TP\in(\mathcal{M})$ it follows that $I+TP=I_{M}\oplus(I_{N}+T_{N})$ is Drazin invertible, and so $I_{N}+T_{N}$ is Drazin invertible by Lemma 2.3 (and hence B-Weyl). Since $T+P=T_{M}\oplus(I_{N}+T_{N})\in{\bf BR}_{i}$, from Lemma 2.3 we obtain that $T_{M}\in{\bf BR}_{i}$. Again by Lemma 2.3 it follows that $T(I-P)+P=T_{M}\oplus I_{N}\in{\bf BR}_{i}$.

(iii)$\Longrightarrow$(i): If (iii) holds, then the closed subspace $M=N(P)$ and $N=R(P)$ define decomposition $(M,N)\in Red(T)$, and since $TP\in(\mathcal{M})$ and $TP=0\oplus T_{N}$, from Lemma 2.2 it follows that $T_{N}\in(\mathcal{M})$. As $T(I-P)+P=T_{M}\oplus I_{N}\in{\bf BR_{i}}$, from Lemma 2.3 it follows that $T_{M}\in{\bf BR}_{i}$. ∎

Theorem 2.3**.**

If $X=H$ is a Hilbert space, or $i=3$ or $6$ or $9$, then the conditions in Theorems 2.2 and 2.1 are equivalent.

Proof.

It is enough to prove that the condition (i) in Theorem 2.2 implies the condition (i) in Theorem 2.1.

Suppose that there exists $(M,N)\in Red(T)$ such that $T_{M}\in{\bf BR}_{i}$ and $T_{N}\in(\mathcal{M})$ and suppose that either $X=H$, or, $i=3$ or $6$ or $9$ (and $X$ is a Banach space). From [4, Theorem 2.7], [6, Lemma 4.1] and [5, Theorem 3.12] it follows that there exists $(M_{1},M_{2})\in Red(T_{M})$ such that $T_{M_{1}}\in{\bf R_{i}}$ and $T_{M_{2}}$ is nilpotent. Define $N_{1}=M_{2}\oplus N$. Then $N_{1}$ is a closed subspace by Lemma 2.4 and $T_{N_{1}}=T_{N}\oplus T_{M_{2}}\in(\mathcal{M})$ by Lemma 2.2. ∎

Definition 2.1**.**

An operator $T\in L(X)$ is meromorphic quasi-polar if there exists a bounded projection $Q$ satisfying

[TABLE]

Theorem 2.4**.**

The following conditions are mutually equivalent for operators $T\in L(X)$:

(i)* There exists $(M,N)\in Red(T)$ such that $T_{M}$ is invertible and $T_{N}\in(\mathcal{M})$;*

(ii)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma(T)$;*

(iii)* $T$ admits a $GK(\mathcal{M})D$ and, $T$ and $T^{\prime}$ have SVEP at 0;*

(iv)* $T$ is generalized Drazin-meromorphic invertible;*

(v)* $T$ is meromorphic quasi-polar;*

(vi)* There exists a projection $P\in L(X)$ such that $P$ commutes with $T$, $TP\in(\mathcal{M})$ and $T+P$ is B-Browder;*

(vii)* There exists a projection $P\in L(X)$ which commutes with $T$ and such that $TP\in(\mathcal{M})$ and $T(I-P)+P$ is B-Browder;*

(viii)* There exists $(M,N)\in Red(T)$ such that $T_{M}$ is B-Browder and $T_{N}\in(\mathcal{M})$;*

(ix)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm acc}\,\sigma_{\bf BB}(T)$;*

(x)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{\bf BB}(T)$.*

Proof.

According to Theorem 2.3, Theorem 2.1, Theorem 2.2 and (1.1) it is sufficient to prove that (ii)$\Longrightarrow$(iii)$\Longrightarrow$ (iv)$\Longrightarrow$(v)$\Longrightarrow$(vi).

If (ii) holds, then $0\notin{\rm int}\,\sigma(T)$ implies that either $0\notin\sigma(T)$ or $0\in\partial\sigma(T)$. In both cases $T$ and $T^{\prime}$ have SVEP at 0; hence (ii) implies (iii).

If (iii) holds, then $(M,N)\in Red(T))$, $T_{N}\in(\mathcal{M})$, $T_{M}$ is Kato and hence, $T^{\prime}_{N^{\bot}}$ is Kato by Lemma 2.6. Since $T$ and $T^{\prime}$ have SVEP at 0, it follows that $T_{M}$ and $T^{\prime}_{N^{\bot}}$ also have SVEP at 0, which implies that $T_{M}$ and $T^{\prime}_{N^{\bot}}$ are injective. As in the proof of [1, Lemma 3.13] it can be proved that $T_{M}$ is surjective, and so $T_{M}$ is invertible. The operator $S=T_{M}^{-1}\oplus 0\in L(M\oplus N)$ satisfies

[TABLE]

Thus (iii) implies (iv).

To prove (iv) implies (v), (assume (iv) and) define the projector $Q\in L(X)$ by setting $ST(=TS)=Q$. Then

[TABLE]

i.e., $T$ is meromorphic quasi-polar.

Assume now that (v) holds. Then there exists a projector $Q\in L(X)$ such that (2.7) holds. Set $P=I-Q$. Then $TP\in(\mathcal{M})$ and for $N=P(X)$ and $M=(I-P)(X)$ we have

[TABLE]

for some $U,V\in L(X)$. Let $U,V\in B(M\oplus N)$have the ($2\times 2$ matrix) representations $U=[U_{ij}]_{i,j=1}^{2}$ and $V=[V_{ij}]_{i,j=1}^{2}$. Then

[TABLE]

and it follows from a straighforward calculation that ($T_{M}$ is invertible, $U_{21}=0=V_{12}$, $U_{12}T_{N}=U_{22}T_{N}=0=T_{N}V_{21}=T_{N}V_{22}$, and hence that) $UTV+P=T_{M}^{-1}\oplus I_{N}$ is invertible with $(UTV+P)^{-1}=T_{M}\oplus I_{N}=T(I-P)+P$. Since $TP\in(\mathcal{M})$, $I+TP$ is Drazin invertible and hence,

[TABLE]

is Drazin invertible, i.e. B-Browder. ∎

The following theorems can be obtained similarly.

Theorem 2.5**.**

The following conditions are mutually equivalent for operators $T\in L(X)$:

(i)* There exists $(M,N)\in Red(T)$ such that $T_{M}$ is bounded below and $T_{N}\in(\mathcal{M})$;*

(ii)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{ap}(T)$;*

(iii)* $T$ admits a $GK(\mathcal{M})D$ and $T$ has SVEP at 0;*

(iv)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm acc}\,\sigma_{\bf BB_{+}}(T)$;*

(v)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{\bf BB_{+}}(T)$.*

Theorem 2.6**.**

The following conditions are mutually equivalent for operators $T\in L(X)$:

(i)* There exists $(M,N)\in Red(T)$ such that $T_{M}$ is surjective and $T_{N}\in(\mathcal{M})$;*

(ii)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{su}(T)$;*

(iii)* $T$ admits a $GK(\mathcal{M})D$ and $T^{\prime}$ has SVEP at 0;*

(iv)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm acc}\,\sigma_{\bf BB_{-}}(T)$;*

(v)* $T$ admits a $GK(\mathcal{M})D$ and $0\notin{\rm int}\,\sigma_{\bf BB_{-}}(T)$.*

We remark that if $T\in L(X)$ is Riesz with infinite spectrum, then $T$ is generalized Drazin-meromorphic invertible, $\sigma(T)=\sigma_{ap}(T)=\sigma_{su}(T)$, $0\in{\rm acc}\,\sigma_{ap}(T)={\rm acc}\,\sigma_{su}(T)$ and $0\notin{\rm int}\,\sigma_{ap}(T)={\rm int}\,\sigma_{su}(T)$. Therefore, the condition that $0\notin{\rm int}\,\sigma_{ap}(T)$ ($0\notin{\rm int}\,\sigma_{su}(T)$) in the statement (ii) in Theorem 2.5 (Theorem 2.6) can not be replaced with the stronger condition that $0\notin{\rm acc}\,\sigma_{ap}(T)$ ($0\notin{\rm acc}\,\sigma_{su}(T)$).

P. Aiena and E. Rosas [2, Theorems 2.2 and 2.5] characterized the SVEP at a point $\lambda_{0}$ in the case that $\lambda_{0}-T$ is of Kato type. Q. Jiang and H. Zhong [3, Theorems 3.5 and 3.9] gave further characterizations of the SVEP at $\lambda_{0}$ in the case that $\lambda_{0}-T$ admits a generalized Kato decomposition. We gave characterizations for the case that $\lambda_{0}-T$ admits a generalized Kato-meromorphic decomposition.

Corollary 2.1**.**

Let $T\in L(X)$ and let $\lambda_{0}-T$ admit a $GK(\mathcal{M})D$. Then the following statements are equivalent:

(i)* $T$ has the SVEP at $\lambda_{0}$;*

(ii)* $\lambda_{0}$ is not an interior point of $\sigma_{ap}(T)$;*

(iii)* $\sigma_{\bf BB_{+}}(T)$ does not cluster at $\lambda_{0}$.*

Proof.

It follows from the equivalences (ii)$\Longleftrightarrow$(iii)$\Longleftrightarrow$(iv) in Theorem 2.5. ∎

Corollary 2.2**.**

Let $T\in L(X)$ and let $\lambda_{0}-T$ admit a $GK(\mathcal{M})D$. Then the following statements are equivalent:

(i)* $T^{\prime}$ has the SVEP at $\lambda_{0}$;*

(ii)* $\lambda_{0}$ is not an interior point of $\sigma_{su}(T)$;*

(iii)* $\sigma_{\bf BB_{-}}(T)$ does not cluster at $\lambda_{0}$.*

Proof.

It follows from Theorem 2.6. ∎

Theorem 2.7**.**

The following statements are equivalent for operator $T\in L(X)$:

(i)* $T=T_{M}\oplus T_{N}$, where $T_{M}$ is invertible, $T_{N}\in(\mathcal{M})$ and $\sigma(T_{N})$ is infinite;*

(ii)* $T$ admits a $GK(\mathcal{M})D$ such that there exists an infinite sequence of poles of resolvent of $T$ converging to [math].*

Proof.

That (i) implies (ii) is a straightforward consequence of the fact that $\sigma(T)=\sigma(T_{M})\cup\sigma(T_{N})$, where $\sigma(T_{N})$ is a coutably infinite set with $\sigma_{D}(T_{N})=\{0\}$.

(ii)$\Longrightarrow$(i): If (ii) holds, then there exists a decomposition $(M,N)\in Red(T)$ such that $T_{M}$ is Kato and $T_{N}\in(\mathcal{M})$. $T_{N}$ being meromorphc, $\sigma(T_{N})$ is either finite or countably infinite (with [math] as its only limit point). We prove that $\sigma(T_{N})$ is finite leads to a contradiction. The hypotheses imply the existence of a sequence $(\mu_{n})$ converging to [math] such that $\mu_{n}\in\sigma(T)$ and $T-\mu_{n}$ is Drazin invertible for all $n\in{\mathbb{N}}$. It means $0\notin{\rm int}\,\sigma_{\bf BB}(T)$ and an argument used to prove that (x) implies (i) in Theorem 2.4 shows that $T_{M}$ is invertible. The spectrum $\sigma(T_{N})$ is finite, $\sigma(T)=\sigma(T_{M})\cup\sigma(T_{N})$ implies that a infinite number of the points $\mu_{n}\in\sigma(T_{M})$, and hence $0\in\sigma(T_{M})$-a contradiction. Hence $\sigma(T_{N})$ is infinite. ∎

Theorem 2.8**.**

If $T\in{\bf gD\mathcal{M}R_{i}}$ and $f$ is holomorphic in a neighbourhood of $\sigma(T)$ such that $f^{-1}(0)\cap\sigma_{\bf R_{i}}(T)=\{0\}$, then $f(T)\in{\bf gD\mathcal{M}R_{i}}$ for all $1\leq i\leq 9$.

Proof.

It is known that $f(\sigma_{\bf R_{i}}(T))=\sigma_{\bf R_{i}}(f(T))$ for all $f$ holomorphic on a neighbourhood of $\sigma(T)$ and $1\leq i\leq 6$. The corresponding inclusion for $7\leq i\leq 9$ is $\sigma_{\bf R_{i}}(f(T))\subset f(\sigma_{\bf R_{i}}(T))$. If $T\in{\bf gD\mathcal{M}R_{i}}$, then there exists a decomposition $(M,N)\in Red(T)$ such that $T_{M}\in{\bf R_{i}}$ and $T_{N}\in(\mathcal{M})$. Furthermore $f(T)=f(T_{M})\oplus f(T_{N})$. Since $f(0)=0$, and since $f$ maps the poles of the resolvent of $T_{N}$ onto the poles of the resolvent of $f(T_{N})$ (see the proof of the first part of [7, Theorem 4.1]), $f(T_{N})\in(\mathcal{M})$. Observe that $0\notin\sigma_{\bf R_{i}}(T_{M})$ and since $f^{-1}(0)\cap\sigma_{\bf R_{i}}(T)=\{0\}$ we conclude that $0\notin f(\sigma_{\bf R_{i}}(T_{M}))$. This, since $f(\sigma_{\bf R_{i}}(T_{M}))\supset\sigma_{\bf R_{i}}(f(T_{M}))$ for all $1\leq i\leq 9$, implies $0\notin\sigma_{\bf R_{i}}(f(T_{M}))$. This completes the proof. ∎

3 Spectra

For $T\in L(X)$, set

[TABLE]

and

[TABLE]

In the following, we shorten $\sigma_{\bf gD\mathcal{M}L(X)^{-1}}(T)$ to

[TABLE]

It is clear from Theorem 2.1 that

[TABLE]

Theorem 3.1**.**

Let $T\in L(X)$ and let $T$ admits a $GK\mathcal{M}D(M,N)$. Then there exists $\epsilon>0$ such that $T-\lambda$ is of Kato type for each $\lambda$ such that $0<|\lambda|<\epsilon$.

Proof.

If $M=\{0\}$, then $T$ is meromorphic and hence $T-\lambda$ is Drazin invertible for all $\lambda\neq 0$. Therefore, $T-\lambda$ is of Kato type for all $\lambda\neq 0$.

Suppose that $M\neq\{0\}$. From [1, Theorem 1.31] it follows that for $|\lambda|<\gamma(T_{M})$, $T_{M}-\lambda$ is Kato. As $T_{N}$ is meromorphic, $T_{N}-\lambda$ is Drazin invertible, and hence it is of Kato type for all $\lambda\neq 0$. Let $\epsilon=\gamma(T_{M})$. According to [14, p. 143] it follows that $T-\lambda$ is of Kato type for each $\lambda$ such that $0<|\lambda|<\epsilon$. ∎

Corollary 3.1**.**

Let $T\in L(X)$. Then

(i)* $\sigma_{\bf gK\mathcal{M}}(T)$ is compact;*

(ii)* The set $\sigma_{Kt}(T)\setminus\sigma_{\bf gK\mathcal{M}}(T)$ consists of at most countably many points.*

Proof.

(i): From Theorem 3.1 it follows that $\sigma_{\bf gK\mathcal{M}}(T)$ is closed and since $\sigma_{\bf gK\mathcal{M}}(T)\subset\sigma(T)$, $\sigma_{\bf gK\mathcal{M}}(T)$ is bounded. Thus $\sigma_{\bf gK\mathcal{M}}(T)$ is compact.

(ii): Suppose that $\lambda_{0}\in\sigma_{Kt}(T)\setminus\sigma_{\bf gK\mathcal{M}}(T)$. Then $T-\lambda_{0}$ admits a $GK\mathcal{M}D$ and according to Theorem 3.1 there exists $\epsilon>0$ such that $T-\lambda$ is of Kato type for each $\lambda$ such that $0<|\lambda-\lambda_{0}|<\epsilon$. This implies that $\lambda_{0}\in{\rm iso}\,\sigma_{Kt}(T)$. Therefore, $\sigma_{Kt}(T)\setminus\sigma_{\bf gK\mathcal{M}}(T)\subset{\rm iso}\,\sigma_{Kt}(T)$, which implies that $\sigma_{Kt}\setminus\sigma_{\bf gK\mathcal{M}}(T)$ is at most countable. ∎

Corollary 3.2**.**

Let $T\in L(X)$ and $1\leq i\leq 9$. Then

(i)* $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset\sigma_{{\bf R}_{i}}(T)$;*

(ii)* $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$ is compact;*

(iii)* ${\rm int}\,\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)={\rm int}\,\sigma_{{\bf R}_{i}}(T)$;*

(iv)* $\partial\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset\partial\sigma_{{\bf R}_{i}}(T)$;*

(v)* $\sigma_{{\bf BR}_{i}}(T)\setminus\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)=({\rm iso}\,\sigma_{{\bf BR}_{i}}(T))\setminus\sigma_{\bf gK\mathcal{M}}(T)$;*

(vi)* The set $\sigma_{{\bf BR}_{i}}(T)\setminus\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$ consist of at most countably many points.*

Proof.

(i) Obvious.

(ii): From (3.2) and Corollary 3.1 (i) it follows that $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$ is closed as the union of two closed sets, while from (i) it follows that $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$ is bounded, and so it is compact.

(iii): From (3.1) we have that ${\rm int}\,\sigma_{{\bf R}_{i}}(T)\subset\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$, and hence ${\rm int}\,\sigma_{{\bf R}_{i}}(T)\subset{\rm int}\,\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$, while from the inclusion (i) it follows that ${\rm int}\,\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset{\rm int}\,\sigma_{{\bf R}_{i}}(T)$. Thus ${\rm int}\,\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)={\rm int}\,\sigma_{{\bf R}_{i}}(T)$.

(iv): Let $\lambda\in\partial\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$. Since $\partial\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset\sigma_{{\bf R}_{i}}(T)$, from $\lambda\notin{\rm int}\,\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)={\rm int}\,\sigma_{{\bf R}_{i}}(T)$ we conclude $\lambda\in\partial\sigma_{{\bf R}_{i}}(T)$. So, $\partial\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)\subset\partial\sigma_{{\bf R}_{i}}(T)$.

(v): It follows from (3.2).

(vi) It follows from (v). ∎

From (3.1) it follows that if $\sigma_{{\bf R}_{i}}(T)$ is countable or contained in a line, then $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)=\sigma_{\bf gK\mathcal{M}}(T)$, $1\leq i\leq 9$.

Corollary 3.3**.**

Let $T\in L(X)$ and $1\leq i\leq 9$. Then

[TABLE]

Proof.

Let $T-\lambda I$ admit a $GK(\mathcal{M})D$ and let $\lambda\in\partial\,\sigma_{{\bf R}_{i}}(T)$. Then $\lambda\notin{\rm int}\,\sigma_{{\bf R}_{i}}(T)$ and according to the equivalence (ii)$\Longleftrightarrow$(iii) in Theorem 2.1 it follows that $\lambda\notin{\rm acc}\,\sigma_{{\bf BR}_{i}}(T)$. Therefore,

[TABLE]

Suppose that $\lambda\in\partial\,\sigma_{{\bf R}_{i}}(T)\cap{\rm acc}\,\sigma_{{\bf BR}_{i}}(T)$. Then there exists a sequence $(\lambda_{n})$ which converges to $\lambda$ and such that $T-\lambda_{n}\in{\bf R_{i}}$ for every $n\in{\mathbb{N}}$. According to [16, Theorem 16.21] it follows that $T-\lambda_{n}$ admits a $GK(\mathcal{M})D$, and so $\lambda_{n}\notin\sigma_{\bf gK\mathcal{M}}(T)$ for every $n\in{\mathbb{N}}$. Since $\lambda\in\sigma_{\bf gK\mathcal{M}}(T)$ by (3.4), we conclude that $\lambda\in\partial\,\sigma_{\bf gK\mathcal{M}}(T)$. This proves the inclusion (3.3). ∎

Similarly to the inclusion (3.4), the following inclusion can be proved

[TABLE]

Corollary 3.4**.**

Let $T\in L(X)$.

(i)* If $T$ has the SVEP, then all accumulation points of $\sigma_{\bf B\mathcal{B}_{+}}(T)$ belong to $\sigma_{\bf gK\mathcal{M}}(T)$.*

(ii)* If $T^{\prime}$ has the SVEP, then all accumulation points of $\sigma_{\bf B\mathcal{B}_{-}}(T)$ belong to $\sigma_{\bf gK\mathcal{M}}(T)$.*

(iii)* If $T$ and $T^{\prime}$ have the SVEP, then all accumulation points of $\sigma_{\bf B\mathcal{B}}(T)$ belong to $\sigma_{\bf gK\mathcal{M}}(T)$.*

Proof.

(i): It follows from the equivalence (iii)$\Longleftrightarrow$(iv) of Theorem 2.5.

(ii): It follows from the equivalence (iii)$\Longleftrightarrow$(iv) of Theorem 2.6.

(iii): It follows from the equivalence (iii)$\Longleftrightarrow$(ix) of Theorem 2.4. ∎

The next corollary extends [1, Corollary 3.118].

Corollary 3.5**.**

Let $T$ be unilateral weighted right shift operator on $\ell_{p}({\mathbb{N}})$, $1\leq p<\infty$, with weight $(\omega_{n})$, and let $c(T)=\lim\limits_{n\to\infty}\inf(\omega_{1}\cdot\dots\cdot\omega_{n})^{1/n}=0$. Then $\sigma_{\bf gK\mathcal{M}}(T)=\sigma_{\bf gD\mathcal{M}{\bf R}_{i}}(T)=\sigma(T)=\overline{D(0,r(T))}$, $1\leq i\leq 9$.

Proof.

According to [1, Corollary 3.118] it follows that $\sigma(T)=\overline{D(0,r(T))}$ and $T$ and $T^{\prime}$ have the SVEP. From the equivalence (ii)$\Longleftrightarrow$(iii) in Theorem 2.4 it follows that $D(0,r(T))={\rm int}\,\sigma(T)\subset\sigma_{\bf gK\mathcal{M}}(T)$. Since $\sigma_{\bf gK\mathcal{M}}(T)$ is closed, we obtain that $\overline{D(0,r(T))}\subset\sigma_{\bf gK\mathcal{M}}(T)\subset\sigma_{\bf gD\mathcal{M}{\bf R}_{i}}(T)\subset\sigma(T)=\overline{D(0,r(T))}$, and so $\sigma_{\bf gK\mathcal{M}}(T)=\sigma_{\bf gD\mathcal{M}{\bf R}_{i}}(T)=\sigma(T)=\overline{D(0,r(T))}$. ∎

The connected hull of a compact subset $K$ of the complex plane ${\mathbb{C}}$, denoted by $\eta K$, is the complement of the unbounded component of ${\mathbb{C}}\setminus K$ [9, Definition 7.10.1].

We recall that, for compact subsets $H,K\subset{\mathbb{C}}$, the following implication holds ([9, Theorem 7.10.3]):

[TABLE]

Evidently, if $K\subseteq{\mathbb{C}}$ is at most countable, then $\eta K=K$. Therefore, for compact subsets $H,K\subseteq{\mathbb{C}}$, if $\eta K=\eta H$, then $H\ {\rm is\ at\ most\ countable}$ if and only if $K\ {\rm is\ at\ most\ countable}$, and in that case $H=K$.

Theorem 3.2**.**

Let $T\in L(X)$. Then

(i)**

[TABLE]

(ii)* $\eta\sigma_{\bf gD\mathcal{M}}(T)=\eta\sigma_{\bf gD\mathcal{M}R_{i}}(T)$, $1\leq i\leq 9$.*

(iii)* The set $\sigma_{\bf gD\mathcal{M}}(T)$ consists of $\sigma_{*}(T)$ and possibly some holes in $\sigma_{*}(T)$ where $\sigma_{*}\in\{\sigma_{\bf gK\mathcal{M}},\break\sigma_{\bf gD\mathcal{M}\mathcal{W}},\sigma_{\bf gD\mathcal{M}\Phi},\sigma_{\bf gD\mathcal{M}\mathcal{W}_{+}},\sigma_{\bf gD\mathcal{M}\Phi_{+}},\sigma_{\bf gD\mathcal{M}\mathcal{J}},\sigma_{\bf gD\mathcal{M}\mathcal{W}_{-}},\sigma_{\bf gD\mathcal{M}\Phi_{-}},\sigma_{\bf gD\mathcal{M}\mathcal{Q}}\}$.*

(iv)* If one of $\sigma_{\bf gK\mathcal{M}}(T)$, $\sigma_{\bf gD\mathcal{M}}(T)$, $\sigma_{\bf gD\mathcal{M}\mathcal{W}}(T)$, $\sigma_{\bf gD\mathcal{M}\Phi}(T)$, $\sigma_{\bf gD\mathcal{M}\mathcal{W}_{+}}(T)$, $\sigma_{\bf gD\mathcal{M}\Phi_{+}}(T)$, $\sigma_{\bf gD\mathcal{M}\mathcal{J}}(T)$, $\sigma_{\bf gD\mathcal{M}\mathcal{W}_{-}}(T)$, $\sigma_{\bf gD\mathcal{M}\Phi_{-}}(T)$, $\sigma_{\bf gD\mathcal{M}\mathcal{Q}}(T)$ is finite (countable), then all of them are equal and hence finite (countable).*

Proof.

Since $\sigma_{\bf gK\mathcal{M}}(T)$ and $\sigma_{\bf gD\mathcal{M}R_{i}}(T)$, $1\leq i\leq 9$, are compact, according to (3.6) and the inclusions

[TABLE]

it is enough to prove that

[TABLE]

Suppose that $\lambda\in\partial\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$. From (3.1) and Corollary 3.2 (iii) it follows that

[TABLE]

Since $\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$ is closed, it follows that $\lambda\in\sigma_{{\bf gD\mathcal{M}R}_{i}}(T)$, and from (3.8) we conclude that $\lambda\in\sigma_{\bf gK\mathcal{M}}(T)$. ∎

Theorem 3.3**.**

Let $T\in L(X)$. The following statements are equivalent:

(i)* $\sigma_{\bf gK\mathcal{M}}(T)=\emptyset$;*

(ii)* $\sigma_{\bf gD\mathcal{M}}(T)=\emptyset$;*

(iii)* $T$ is polynomially meromorphic;*

(iv)* $\sigma_{\bf B\mathcal{B}}(T)$ is a finite set.*

Proof.

The equivalence (i)$\Longleftrightarrow$(ii) follows from Theorem 3.2.

The equivalence (iii)$\Longleftrightarrow$(iv) has been proved in [7].

(ii) $\Longrightarrow$ (iv): Suppose that $\sigma_{\bf gD\mathcal{M}}(T)=\emptyset$. From Theorem 2.4, (iv) $\Longleftrightarrow$ (ix), it follows that $\sigma_{\bf gD\mathcal{M}}(T)=\sigma_{\bf gK\mathcal{M}}(T)\cup{\rm acc}\,\sigma_{\bf B\mathcal{B}}(T)$, and so ${\rm acc}\,\sigma_{\bf B\mathcal{B}}(T)=\emptyset$ which implies that $\sigma_{\bf B\mathcal{B}}(T)$ is a finite set.

(iii) $\Longrightarrow$ (ii): Let $T$ be polynomially meromorphic and $p^{-1}(0)=\{\lambda_{1},\dots,\lambda_{n}\}$ where $p(T)\in(\mathcal{M})$. According to [7, Corollary 4.3] we have that $\sigma_{\bf B\mathcal{B}}(T)\subset p^{-1}(0)$. It implies that $T-\lambda$ is Drazin invertible and hence, generalized Drazin-meromorphic invertible for every $\lambda\notin p^{-1}(0)$. According to [7, Theorem 4.9], $X$ is decomposed into the direct sum $X=X_{1}\oplus\dots\oplus X_{n}$ where $X_{i}$ is closed $T$-invariant subspace of $X$, $T=T_{1}\oplus\dots\oplus T_{n}$ where $T_{i}$ is the reduction of $T$ on $X_{i}$ and $T_{i}-\lambda_{i}\in(\mathcal{M})$, $i=1,\dots,n$. From $T_{i}-\lambda_{i}\in(\mathcal{M})$, it follows that $\sigma_{\bf B\mathcal{B}}(T_{i}-\lambda_{i})\subset\{0\}$ and hence, $\sigma_{\bf B\mathcal{B}}(T_{i})\subset\{\lambda_{i}\}$, $i=1,\dots,n$. It implies that $T_{i}-\lambda_{j}$ is B-Browder for $i\neq j$, $i,j\in\{1,\dots,n\}$.

Consider the decomposition

[TABLE]

From Lemma 2.4 it follows that $X_{2}\oplus\dots\oplus X_{n}$ is closed. Since $(X_{1},X_{2}\oplus\dots\oplus X_{n})\in Red(T)$, $(T-\lambda_{1})_{X_{1}}=T_{1}-\lambda_{1}\in(\mathcal{M})$, and since $(T-\lambda_{1})_{X_{2}\oplus\dots\oplus X_{n}}=(T_{2}-\lambda_{1})\oplus\dots\oplus(T_{n}-\lambda_{1})$ is B-Browder as a direct sum of B-Browder operators $T_{2}-\lambda_{1},\dots,T_{n}-\lambda_{1}$, it follows that $T-\lambda_{1}$ is generalized Drazin-meromorphic invertible. In that way we can prove that $T-\lambda_{i}$ is generalized Drazin-meromorphic invertible for every $i\in\{1,\dots,n\}$. Therefore, $T-\lambda$ is generalized Drazin-meromorphic invertible for every $\lambda\in{\mathbb{C}}$, and so $\sigma_{\bf gD\mathcal{M}}(T)=\emptyset$. ∎

P. Aiena and E. Rosas [2, Theorem 2.10] proved that if $T\in L(X)$ be an operator for which $\sigma_{ap}(T)=\partial\sigma(T)$ and every $\lambda\in\partial\sigma(T)$ is not isolated in $\sigma(T)$, then $\sigma_{ap}(T)=\sigma_{Kt}(T)$, while Q. Jiang and H. Zhong [3, Theorem 3.12] improved this result by proving that under the same conditions it holds $\sigma_{ap}(T)=\sigma_{gK}(T)$. In [20, Theorem 3.14] it was proved that $\sigma_{ap}(T)=\sigma_{gKR}(T)$. The next theorem improves these results.

Theorem 3.4**.**

For $T\in L(X)$ suppose that $\sigma_{ap}(T)=\partial\sigma(T)$ and every $\lambda\in\partial\sigma(T)$ is not isolated in $\sigma(T)$. Then

[TABLE]

Proof.

From the proof of [20, Theorem 3.14] we have $\sigma_{ap}(T)={\rm acc}\,\sigma_{ap}(T)=\partial\,\sigma_{ap}(T)$. According to the inclusion (3.4) it holds

[TABLE]

From [21, Corollary 4.9 (i)] we have that $\sigma_{ap}(T)=\sigma_{TUD}(T)$, and since $\sigma_{TUD}(T)\subset\sigma_{\bf B\mathcal{B}_{+}}(T)\subset\sigma_{ap}(T)$ [5], it follows that $\sigma_{\bf B\mathcal{B}_{+}}(T)=\sigma_{ap}(T)$. Hence $\partial\,\sigma_{ap}(T)\cap{\rm acc}\,\sigma_{\bf B\mathcal{B}_{+}}(T)=\sigma_{ap}(T)$, which together with (3.9) gives $\sigma_{ap}(T)\subset\sigma_{\bf gK\mathcal{M}}(T)$. As $\sigma_{\bf gK\mathcal{M}}(T)\subset\sigma_{ap}(T)$, we get that $\sigma_{ap}(T)=\sigma_{\bf gK\mathcal{M}}(T)$. ∎

Theorem 3.5**.**

Let $T\in L(X)$ be an operator for which $\sigma_{su}(T)=\partial\sigma(T)$ and every $\lambda\in\partial\sigma(T)$ is not isolated in $\sigma(T)$. Then

[TABLE]

Proof.

Follows from [21, Corollary 4.9 (ii)] analogously to the proof of Theorem 3.4. ∎

Using Theorems 3.4 and 3.5 we find the generalized Kato-meromorphic spectra and $gD\mathcal{M}R_{i}$-spectra, $1\leq i\leq 9$, for some operators.

Example 3.6**.**

For the $Ces\acute{a}ro\ operator$ $C_{p}$ defined on the classical Hardy space $H_{p}(\bf{D})$, $\bf{D}$ the open unit disc and $1<p<\infty$, by

[TABLE]

it is known that its spectrum is the closed disc $\Gamma_{p}$ centered at $p/2$ with radius $p/2$, $\sigma_{gKR}(C_{p})=\sigma_{gK}(C_{p})=\sigma_{Kt}(C_{p})=\sigma_{ap}(C_{p})=\partial\Gamma_{p}$ and also $\sigma_{\Phi}(C_{p})=\partial\Gamma_{p}$ [15], [2], [20]. From Theorem 3.4 it follows that $\sigma_{\bf gK\mathcal{M}}(C_{p})=\sigma_{\bf gD\mathcal{M}\Phi_{+}}(C_{p})=\sigma_{\bf gD\mathcal{M}\mathcal{W}_{+}}(C_{p})=\sigma_{\bf gD\mathcal{M}\mathcal{J}}(C_{p})=\sigma_{ap}(C_{p})=\partial\Gamma_{p}$, and since ${\rm int}\,\sigma_{\Phi}(C_{p})={\rm int}\,\sigma_{\Phi_{-}}(C_{p})=\emptyset$, according to (3.1) we have that $\sigma_{\bf gD\mathcal{M}\Phi}(C_{p})=\sigma_{\bf gD\mathcal{M}\Phi_{-}}(C_{p})=\sigma_{\bf gK\mathcal{M}}(C_{p})=\partial\Gamma_{p}.$ From $\sigma(C_{p})=\Gamma_{p}$ and $\sigma_{ap}(C_{p})=\partial\Gamma_{p}$ it follows that and $\sigma_{su}(C_{p})=\Gamma_{p}$ which together with $\sigma_{\Phi}(C_{p})=\partial\Gamma_{p}$ implies that $\sigma_{\mathcal{W}_{-}}(C_{p})=\sigma_{\mathcal{W}}(C_{p})=\Gamma_{p}$. Again from (3.1) we conclude that $\sigma_{\bf gD\mathcal{M}\mathcal{W}_{-}}(C_{p})=\sigma_{\bf gD\mathcal{M}\mathcal{W}}(C_{p})=\sigma_{\bf gD\mathcal{M}\mathcal{Q}}(C_{p})=\sigma_{\bf gD\mathcal{M}}(C_{p})=\Gamma_{p}$. **

Example 3.7**.**

For each $X\in\{c_{0}({\mathbb{N}}),c({\mathbb{N}}),\ell_{\infty}({\mathbb{N}}),\ell_{p}({\mathbb{N}})\}$, $p\geq 1$, and the forward and backward unilateral shifts $U$, $V\in L(X)$ there are equalities $\sigma(U)=\sigma(V)=\mathbb{D}$, $\sigma_{D}(U)=\sigma_{D}(V)=\mathbb{D}$, $\sigma_{ap}(U)=\sigma_{su}(V)=\partial\mathbb{D}$, $\sigma_{\Phi}(U)=\sigma_{\Phi}(V)=\partial\mathbb{D}$, where $\mathbb{D}=\{\lambda\in{\mathbb{C}}:|\lambda|\leq 1\}$ [19, Theorem 4.2]. As in the previous example, from Theorem 3.4 we conclude that $\sigma_{\bf gK\mathcal{M}}(U)=\sigma_{ap}(U)=\partial\mathbb{D}$, and hence

[TABLE]

while from (3.1) we get $\sigma_{\bf gD\mathcal{M}\mathcal{W}_{-}}(U)=\sigma_{\bf gD\mathcal{M}\mathcal{W}}(U)=\sigma_{\bf gD\mathcal{M}\mathcal{Q}}(U)=\sigma_{\bf gD\mathcal{M}}(U)=\mathbb{D}$.

From Theorem 3.5 it follows that

[TABLE]

As $\sigma_{\Phi}(V)=\partial\mathbb{D}$, from (3.1) we get that $\sigma_{\bf gD\mathcal{M}\Phi}(V)=\sigma_{\bf gD\mathcal{M}\Phi_{+}}(V)=\partial\mathbb{D}$. From $\sigma_{\Phi}(V)=\partial\mathbb{D}$, $\sigma_{ap}(V)=\mathbb{D}$ and $\sigma_{su}(V)=\partial\mathbb{D}$, we conclude that for $|\lambda|<1$ it holds that $V-\lambda I$ is Fredholm with positive index and so, $\{\lambda\in{\mathbb{C}}:|\lambda|<1\}\subset\sigma_{\mathcal{W}_{+}}(V)\subset\sigma_{\mathcal{W}}(V)\subset\mathbb{D}$, which implies that $\sigma_{\mathcal{W}_{+}}(V)=\sigma_{\mathcal{W}}(V)=\mathbb{D}$. Now again from (3.1) it follows that $\sigma_{\bf gD\mathcal{M}\mathcal{W}_{+}}(V)=\sigma_{\bf gD\mathcal{M}\mathcal{W}}(V)=\mathbb{D}$, and hence $\sigma_{\bf gD\mathcal{M}\mathcal{J}}(V)=\mathbb{D}$ and $\sigma_{\bf gD\mathcal{M}}(V)=\mathbb{D}$.

Every non-invertible isometry $T$ has the property that $\sigma(T)=\mathbb{D}$ and $\sigma_{ap}(T)=\partial\mathbb{D}$ [2, p. 187], and hence $\sigma_{ap}(T)=\partial\sigma(T)$ and every $\lambda\in\partial\sigma(T)$ is not isolated in $\sigma(T)$. By Theorem 3.4 and (3.1) it follows that $\sigma_{\bf gK\mathcal{M}}(T)=\sigma_{\bf gD\mathcal{M}\Phi_{+}}(T)=\sigma_{\bf gD\mathcal{M}\mathcal{W}_{+}}(T)=\sigma_{\bf gD\mathcal{M}\mathcal{J}}(T)=\partial\mathbb{D}$ and $\sigma_{\bf gD\mathcal{M}\mathcal{Q}}(T)=\mathbb{D}$.