An indefinite range inclusion theorem for triplets of bounded linear operators on a Hilbert space

TL;DR

This paper investigates triplets of bounded linear operators on a Hilbert space, establishing a range inclusion theorem with norm estimates using Kren space geometry and de Branges-Rovnyak space theory.

Contribution

It introduces a new range inclusion theorem for operator triplets, combining Kren space geometry and de Branges-Rovnyak space theory to derive norm estimates.

Findings

Established a range inclusion theorem with norm bounds

Applied Kren space geometry in operator analysis

Utilized de Branges-Rovnyak space theory for operator inequalities

Abstract

We study triplets of Hilbert space operators satisfying a certain inequality. A range inclusion theorem with norm estimate for those triplets is given with the language of Kre\u{\i}n space geometry and de Branges-Rovnyak space theory.

Equations

$$0\le T_1{T}_1^{\ast }+T_2{T}_2^{\ast }-T_3{T}_3^{\ast }\le I.$$

$$T=(T_1{T}_1^{\ast }+T_2{T}_2^{\ast }-T_3{T}_3^{\ast }{)}^{1/2}.$$

$$\Vert \left(\begin{array}{cc}{T}_{{\phi }_1}& T_{{\phi }_2}\end{array}\right)\Vert \le 1.$$

$$0\le T_{{\phi }_1}{T}_{{\phi }_1}^{\ast }+T_{{\phi }_2}{T}_{{\phi }_2}^{\ast }\le I.$$

$$\Vert \left(\begin{array}{c}{T}_{{\psi }_1}\\ T_{{\psi }_2}\end{array}\right)\Vert \le 1.$$

$${\phi }_3={\phi }_1{\psi }_1+{\phi }_2{\psi }_2=\left(\begin{array}{cc}{\phi }_1& {\phi }_2\end{array}\right)\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\end{array}\right),$$

$$0\le T_{{\phi }_1}{T}_{{\phi }_1}^{\ast }+T_{{\phi }_2}{T}_{{\phi }_2}^{\ast }-T_{{\phi }_3}{T}_{{\phi }_3}^{\ast }\le T_{{\phi }_1}{T}_{{\phi }_1}^{\ast }+T_{{\phi }_2}{T}_{{\phi }_2}^{\ast }\le I.$$

$$T_z{T}_z^{\ast }+T_w{T}_w^{\ast }-T_{zw}{T}_{zw}^{\ast }=T_z{T}_z^{\ast }+T_w(I-T_z{T}_z^{\ast })T_w^{\ast },$$

$$T_{{\phi }_1}{T}_{{\phi }_1}^{\ast }+T_{{\phi }_2}{T}_{{\phi }_2}^{\ast }-T_{{\phi }_3}{T}_{{\phi }_3}^{\ast }=P_{\mathcal{M}},$$

$$T_{{\phi }_1}^{\ast }{T}_{{\phi }_1}+T_{{\phi }_2}^{\ast }{T}_{{\phi }_2}-T_{{\phi }_3}^{\ast }{T}_{{\phi }_3}=I.$$

$${\mathcal{H}}_{+}=\mathcal{H}\oplus \mathcal{H},{\mathcal{H}}_{-}=\mathcal{H}\text{mbox}andJ=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right),$$

$$⟨\mathbf{x},\mathbf{y}{⟩}_{\mathcal{K}}=⟨J\mathbf{x},\mathbf{y}{⟩}_{\mathcal{H}\oplus \mathcal{H}\oplus \mathcal{H}}=⟨x_1,y_1{⟩}_{\mathcal{H}}+⟨x_2,y_2{⟩}_{\mathcal{H}}-⟨x_3,y_3{⟩}_{\mathcal{H}}.$$

$$\mathbb{T}:\mathcal{K}\to \mathcal{H},\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)\mapsto T_1{x}_1+T_2{x}_2-T_3{x}_3.$$

$$⟨x,\mathbb{T}(x_1,x_2,x_3{)}^t{⟩}_{\mathcal{H}}$$

$$=⟨T_1^{\ast }x,x_1{⟩}_{\mathcal{H}}+⟨T_2^{\ast }x,x_2{⟩}_{\mathcal{H}}-⟨T_3^{\ast }x,x_3{⟩}_{\mathcal{H}}$$

$$=⟨(T_1^{\ast }x,T_2^{\ast }x,T_3^{\ast }x{)}^t,(x_1,x_2,x_3{)}^t{⟩}_{\mathcal{K}},$$

$${\mathbb{T}}^{♯}:\mathcal{H}\to \mathcal{K},x\mapsto \left(\begin{array}{c}{T}_1^{\ast }x\\ T_2^{\ast }x\\ T_3^{\ast }x\end{array}\right).$$

$$\mathbb{T}{\mathbb{T}}^{♯}x=T_1{T}_1^{\ast }x+T_2{T}_2^{\ast }x-T_3{T}_3^{\ast }x.$$

$$T=(T_1{T}_1^{\ast }+T_2{T}_2^{\ast }-T_3{T}_3^{\ast }{)}^{1/2}.$$

$$VTx=\left(\begin{array}{c}{T}_1^{\ast }x\\ T_2^{\ast }x\\ T_3^{\ast }x\end{array}\right)(x\in (\ker T{)}^{\perp }).$$

$$\Vert Tx{\Vert }_{\mathcal{H}}^2=\Vert T_1^{\ast }x{\Vert }_{\mathcal{H}}^2+\Vert T_2^{\ast }x{\Vert }_{\mathcal{H}}^2-\Vert T_3^{\ast }x{\Vert }_{\mathcal{H}}^2=⟨{\mathbb{T}}^{♯}x,{\mathbb{T}}^{♯}x{⟩}_{\mathcal{K}}$$

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$$\Vert \mathbf{x}{\Vert }_{{\mathcal{K}}_0}^2=⟨\mathbf{x},\mathbf{x}{⟩}_{{\mathcal{K}}_0}=\underset{n\to \infty }{\lim }⟨\mathbf{x},{\mathbb{T}}^{♯}{w}_n{⟩}_{{\mathcal{K}}_0}=\underset{n\to \infty }{\lim }⟨\tilde{\mathbb{T}}\mathbf{x},w_n{⟩}_{\mathcal{H}}=0.$$

$$⟨\mathbb{T}{\mathbb{T}}^{♯}x,y{⟩}_{\mathcal{H}}=⟨{\mathbb{T}}^{♯}x,{\mathbb{T}}^{♯}y{⟩}_{\mathcal{K}}=⟨{\mathbb{T}}^{♯}x,{\mathbb{T}}^{♯}y{⟩}_{{\mathcal{K}}_0}=⟨\tilde{\mathbb{T}}{\mathbb{T}}^{♯}x,y{⟩}_{\mathcal{H}}(x,y\in \mathcal{L}),$$

$$\mathcal{L}\supset \tilde{\mathbb{T}}{\mathcal{K}}_0\supset \mathbb{T}{\mathbb{T}}^{♯}\mathcal{L}=\mathrm{ran}{T}^2.$$

$$\Vert Tx{\Vert }_{\mathcal{M}(T)}=\Vert P_{(\ker T{)}^{\perp }}x{\Vert }_{\mathcal{H}}=\min \{\Vert y{\Vert }_{\mathcal{H}}:Ty=Tx\},$$

$$T=(T_1{T}_1^{\ast }+T_2{T}_2^{\ast }-T_3{T}_3^{\ast }{)}^{1/2}.$$

$$E_{\lambda }^{\perp }=I_{\mathcal{H}}-E_{\lambda }=E((\lambda ,\infty )).$$

$$x_{\epsilon }=E_{\epsilon }^{\perp }x,y_{\epsilon }=\left({\int }_{\epsilon }^{\infty }\frac{1}{\lambda }d{E}_{\lambda }\right)x_{\epsilon }\text{mbox}and{\mathbf{z}}_{\epsilon }={\mathbb{T}}^{♯}{y}_{\epsilon }.$$

$$\mathbb{T}{\mathbf{z}}_{\epsilon }=\mathbb{T}{\mathbb{T}}^{♯}{y}_{\epsilon }=T^2{y}_{\epsilon }=T^2\left({\int }_{\epsilon }^{\infty }\frac{1}{\lambda }d{E}_{\lambda }\right)x_{\epsilon }=T{x}_{\epsilon }.$$

$$\Vert u-\mathbb{T}{\mathbf{z}}_{\epsilon }{\Vert }_{\mathcal{H}}=\Vert Tx-T{x}_{\epsilon }{\Vert }_{\mathcal{H}}\le \Vert x-x_{\epsilon }{\Vert }_{\mathcal{H}}\to 0(\epsilon \to 0).$$

Full text

2010 Mathematical Subject Classification: Primary 47B50; Secondary 47B32

keywords: Kreĭn space, de Branges-Rovnyak space, Toeplitz operator

1 Introduction

Let $T_{1}$, $T_{2}$ and $T_{3}$ be bounded linear operators on a Hilbert space ${\mathcal{H}}$. In this paper, we are going to study triplet $(T_{1},T_{2},T_{3})$ satisfying the following inequality:

[TABLE]

Let ${\mathfrak{T}}({\mathcal{H}})$ denote the set of operator triplets satisfying (1.1) on ${\mathcal{H}}$. For any triplet $(T_{1},T_{2},T_{3})$ in ${\mathfrak{T}}({\mathcal{H}})$, we set

[TABLE]

As the main theorem of this paper, we will show the following: for any vector $u$ in $\operatorname{ran}T$, there exists some vector $\mathbf{z}_{\varepsilon}=(z_{1}(\varepsilon),z_{2}(\varepsilon),z_{3}(\varepsilon))^{t}$ in ${\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}$ such that

1. (i)

$T_{1}z_{1}(\varepsilon)+T_{2}z_{2}(\varepsilon)-T_{3}z_{3}(\varepsilon)\to u$ $(\varepsilon\to 0)$ in the strong topology of ${\mathcal{H}}$, 2. (ii)

$0\leq\|z_{1}(\varepsilon)\|_{{\mathcal{H}}}^{2}+\|z_{2}(\varepsilon)\|_{{\mathcal{H}}}^{2}-\|z_{3}(\varepsilon)\|_{{\mathcal{H}}}^{2}\uparrow\|u\|_{{\mathcal{M}}(T)}^{2}$ $(\varepsilon\downarrow 0)$,

where $\|u\|_{{\mathcal{M}}(T)}$ denotes the norm of $u$ in the de Branges-Rovnyak space induced by $T$.

This paper is organized as follows. In Section 2, by giving examples from operator theory on Hardy spaces, it is shown that ${\mathfrak{T}}({\mathcal{H}})$ is nontrivial. In Section 3, we study indefinite inner product spaces induced by triplets in ${\mathfrak{T}}({\mathcal{H}})$, and prove the main theorem (Theorem 3.1). In Section 4, we investigate the local structure of range spaces of operators appearing in Section 3.

2 Examples

Trivial examples of triplets in ${\mathfrak{T}}({\mathcal{H}})$ are easily obtained from Douglas’ range inclusion theorem. We shall see that ${\mathfrak{T}}({\mathcal{H}})$ is nontrivial.

Example 2.1**.**

Let $H^{2}$ be the Hardy space over the unit disk, and let $H^{\infty}$ be the Banach algebra consisting of all bounded analytic functions in $H^{2}$. For any function $\varphi$ in $H^{\infty}$, $T_{\varphi}$ denotes the Toeplitz operator with symbol $\varphi$. We choose $\varphi_{1}$ and $\varphi_{2}$ from $H^{\infty}$ satisfying

[TABLE]

Then this norm inequality is equivalent to that

[TABLE]

Further, we choose $\psi_{1}$ and $\psi_{2}$ from $H^{\infty}$ satisfying

[TABLE]

Then, setting

[TABLE]

by the generalized Toeplitz-corona theorem (see Theorem 8.57 in Agler-McCarthy [1]), we have that

[TABLE]

Our study has been motivated by the next example.

Example 2.2** (Wu-Seto-Yang [5]).**

Let $H^{2}$ be the Hardy space over the unit disk. Then the tensor product Hilbert space $H^{2}\otimes H^{2}$ is isomorphic to the Hardy space over the bidisk. Let $z$ and $w$ denote coordinate functions, and let $T_{z}$ and $T_{w}$ be Toeplitz operators with symbols $z$ and $w$, respectively. We note that $T_{z}$ and $T_{w}$ are doubly commuting isometries on $H^{2}\otimes H^{2}$. In fact, $T_{z}$ and $T_{w}$ are identified with $T_{z}\otimes I$ and $I\otimes T_{w}$, respectively. Now, since orthogonal projections $T_{z}T_{z}^{\ast}$ and $T_{w}(I-T_{z}T_{z}^{\ast})T_{w}^{\ast}$ are commuting,

[TABLE]

is the orthogonal projection onto $(H^{2}\otimes H^{2})\ominus{\mathbb{C}}$. Hence $(T_{z},T_{w},T_{zw})$ belongs to ${\mathfrak{T}}(H^{2}\otimes H^{2})$. Further non-trivial examples can be obtained from the module structure of $H^{2}\otimes H^{2}$. Let ${\mathcal{M}}$ be a closed subspace of $H^{2}\otimes H^{2}$. Then ${\mathcal{M}}$ is called a submodule if ${\mathcal{M}}$ is invariant for $T_{z}$ and $T_{w}$. For many examples of submodules in $H^{2}\otimes H^{2}$, there exist bounded analytic functions $\varphi_{1}$, $\varphi_{2}$ and $\varphi_{3}$ on the bidisk such that

[TABLE]

where $P_{{\mathcal{M}}}$ denotes the orthogonal projection onto ${\mathcal{M}}$, and

[TABLE]

Remark 2.1**.**

We note that Example 2.1 is deduced from the complete Pick property. On the other hand, the kernel of $H^{2}\otimes H^{2}$ does not have it.

3 Indefinite range inclusion

Setting

[TABLE]

we consider the Kreĭn space ${\mathcal{K}}=({\mathcal{H}}_{+}\oplus{\mathcal{H}}_{-},J)$, that is, for any vectors $\mathbf{x}=(x_{1},x_{2},x_{3})^{t}$ and $\mathbf{y}=(y_{1},y_{2},y_{3})^{t}$ in ${\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}$, the inner product of $\mathbf{x}$ and $\mathbf{y}$ in ${\mathcal{K}}$ is defined to be

[TABLE]

For basic Kreĭn space geometry, see Dritschel-Rovnyak [3].

Let $(T_{1},T_{2},T_{3})$ be a triplet in ${\mathfrak{T}}({\mathcal{H}})$. Then we define a linear operator ${\mathbb{T}}$ as follows:

[TABLE]

The adjoint operator ${\mathbb{T}}^{\sharp}$ of ${\mathbb{T}}$ with respect to inner products of ${\mathcal{H}}$ and ${\mathcal{K}}$ is obtained as follows:

[TABLE]

that is, we have that

[TABLE]

In particular, we have that

[TABLE]

For any $(T_{1},T_{2},T_{3})$ in ${\mathfrak{T}}({\mathcal{H}})$, we set

[TABLE]

Note that $T$ is positive and contractive. Consider the operator $V:\operatorname{ran}T\to{\mathbb{T}}^{\sharp}(\ker T)^{\perp}$ defined by

[TABLE]

Then it follows from the identity

[TABLE]

that $V$ is an isometry and ${\mathbb{T}}^{\sharp}(\ker T)^{\perp}$ is a pre-Hilbert space. Let ${\mathcal{K}}_{0}$ be the completion of ${\mathbb{T}}^{\sharp}(\ker T)^{\perp}$ with the norm induced by (3.1), and let $\widetilde{V}:\overline{\operatorname{ran}}T\to{\mathcal{K}}_{0}$ denote the isometric extension of $V$, in fact, $\widetilde{V}$ is unitary. Then ${\mathbb{T}}^{\sharp}=\widetilde{V}T$ on $(\ker T)^{\perp}$ gives the polar decomposition of ${\mathbb{T}}^{\sharp}$, that is,

[TABLE]

is commutative.

Further, it follows from (3.1) that ${\mathbb{T}}^{\sharp}$ is bounded in (3.2). Hence, we can take the Hilbert space adjoint $\widetilde{{\mathbb{T}}}$ of ${\mathbb{T}}^{\sharp}$ in (3.2). We summarize basic properties of $\widetilde{{\mathbb{T}}}$ in the following proposition:

Proposition 3.1**.**

Let $\widetilde{{\mathbb{T}}}:{\mathcal{K}}_{0}\to{\mathcal{L}}$ be the Hilbert space adjoint of ${\mathbb{T}}^{\sharp}$ in the sense of (3.2). Then

1. (i)

$\widetilde{{\mathbb{T}}}$* is injective,* 2. (ii)

$\widetilde{{\mathbb{T}}}$* is the extension of ${\mathbb{T}}|_{{\mathbb{T}}^{\sharp}{\mathcal{L}}}$,* 3. (iii)

$\operatorname{ran}\widetilde{{\mathbb{T}}}$* is dense in ${\mathcal{L}}$.*

Proof.

Suppose that $\widetilde{{\mathbb{T}}}\mathbf{x}=0$ for some $\mathbf{x}$ in ${\mathcal{K}}_{0}$. Then there exists a sequence $\{w_{n}\}_{n}$ in ${\mathcal{L}}=(\ker T)^{\perp}$ such that $\|\mathbf{x}-{\mathbb{T}}^{\sharp}w_{n}\|_{{\mathcal{K}}_{0}}\to 0$ as $n\to\infty$. Hence we have that

[TABLE]

Thus we have (i). Further, since

[TABLE]

we have (ii). It follows from (ii) that

[TABLE]

This concludes (iii). ∎

Let ${\mathcal{M}}(T)$ denote the de Branges-Rovnyak space induced by $T$, that is, ${\mathcal{M}}(T)$ is the Hilbert space consisting of all vectors in $\operatorname{ran}T$ with the pull-back norm

[TABLE]

where $P_{(\ker T)^{\perp}}$ is the orthogonal projection from ${\mathcal{H}}$ onto $(\ker T)^{\perp}$.

Theorem 3.1**.**

Let ${\mathcal{H}}$ be a Hilbert space. For any triplet $(T_{1},T_{2},T_{3})$ in ${\mathfrak{T}}({\mathcal{H}})$, we set

[TABLE]

If $u$ belongs to ${\mathcal{M}}(T)$, then, for any $\varepsilon>0$, there exists some vector $\mathbf{z}_{\varepsilon}=(z_{1}(\varepsilon),z_{2}(\varepsilon),z_{3}(\varepsilon))^{t}$ in ${\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}$ such that

1. (i)

$T_{1}z_{1}(\varepsilon)+T_{2}z_{2}(\varepsilon)-T_{3}z_{3}(\varepsilon)\to u$* $(\varepsilon\to 0)$ in the strong topology of ${\mathcal{H}}$,* 2. (ii)

$0\leq\|z_{1}(\varepsilon)\|_{{\mathcal{H}}}^{2}+\|z_{2}(\varepsilon)\|_{{\mathcal{H}}}^{2}-\|z_{3}(\varepsilon)\|_{{\mathcal{H}}}^{2}\uparrow\|u\|_{{\mathcal{M}}(T)}^{2}$* $(\varepsilon\downarrow 0)$,* 3. (iii)

$\mathbf{z}_{\varepsilon}$* converges to some vector $\mathbf{z}$ in the strong topology of ${\mathcal{K}}_{0}$ and $u=\widetilde{{\mathbb{T}}}\mathbf{z}$.*

Proof.

Let $\{E_{\lambda}\}_{\lambda\in{\mathbb{R}}}$ be the spectral family of $T$, and we set

[TABLE]

Suppose that $u=Tx$ where $x$ is in ${\mathcal{L}}=(\ker T)^{\perp}$. Then, for arbitrary $\varepsilon>0$, put

[TABLE]

We note that $y_{\varepsilon}$ and $\mathbf{z}_{\varepsilon}$ belong to ${\mathcal{L}}$ and ${\mathcal{K}}_{0}$, respectively. Then we have that

[TABLE]

Hence we have that

[TABLE]

Thus we have (i). Further, it follows from $Ty_{\varepsilon}=x_{\varepsilon}$ that

[TABLE]

This concludes (ii). Finally, since

[TABLE]

$\mathbf{z}_{\varepsilon}$ converges to some vector $\mathbf{z}$ in ${\mathcal{K}}_{0}$, and

[TABLE]

Thus we have (iii). ∎

Corollary 3.1**.**

Suppose that ${\mathcal{H}}$ is of finite dimension. If $u$ belongs to ${\mathcal{M}}(T)$, then there exists some $\mathbf{z}=(z_{1},z_{2},z_{3})^{t}$ in ${\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}$ such that

[TABLE]

and

[TABLE]

Proof.

If $\varepsilon>0$ is sufficiently small, then $E_{\varepsilon}^{\perp}=P_{(\ker T)^{\perp}}$. ∎

Theorem 3.1 can be generalized to any finite operator tuple $(T_{1},\ldots,T_{m},T_{m+1},\ldots,T_{n})$ satisfying

[TABLE]

In particular, the proof of Theorem 3.1 can be applied to the de Branges-Rovnyak complement ${\mathcal{H}}(T)={\mathcal{M}}(\sqrt{I-TT^{\ast}})$ of ${\mathcal{M}}(T)$.

Theorem 3.2**.**

Let ${\mathcal{H}}$ be a Hilbert space. For any triplet $(T_{1},T_{2},T_{3})$ in ${\mathfrak{T}}({\mathcal{H}})$, we set

[TABLE]

Then, for any $v$ in ${\mathcal{H}}(T)$ and $\varepsilon>0$, there exists some vector $\mathbf{w}_{\varepsilon}=(w_{0}(\varepsilon),w_{1}(\varepsilon),w_{2}(\varepsilon),w_{3}(\varepsilon))^{t}$ in ${\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}\oplus{\mathcal{H}}$ such that

1. (i)

$w_{0}(\varepsilon)-T_{1}w_{1}(\varepsilon)-T_{2}w_{2}(\varepsilon)+T_{3}w_{3}(\varepsilon)\to v$ $(\varepsilon\to 0)$ in the strong topology of ${\mathcal{H}}$, 2. (ii)

$0\leq\|w_{0}(\varepsilon)\|_{{\mathcal{H}}}^{2}-\|w_{1}(\varepsilon)\|_{{\mathcal{H}}}^{2}-\|w_{2}(\varepsilon)\|_{{\mathcal{H}}}^{2}+\|w_{3}(\varepsilon)\|_{{\mathcal{H}}}^{2}\uparrow\|v\|_{{\mathcal{H}}(T)}^{2}$ $(\varepsilon\downarrow 0)$.

In the proof of Theorem 3.1, we essentially showed that ${\mathcal{M}}(T)$ is contractively embedded into ${\mathcal{M}}(\widetilde{{\mathbb{T}}})$. Moreover, applying the same method in the proof of Theorem 4.3 stated in the next section, we can conclude that the converse is also true, that is, ${\mathcal{M}}(T)={\mathcal{M}}(\widetilde{{\mathbb{T}}})$ as Hilbert spaces. However, ${\mathcal{K}}_{0}$ and $\widetilde{{\mathbb{T}}}$ seem to be rather elusive obejects. Thus, in the next section, we will investigate the local structure of range spaces of $T$ and ${\mathbb{T}}$.

4 Local structure of range spaces

In this section, we need some facts from de Branges-Rovnyak space theory and Kreĭn space geometry. Let ${\mathcal{H}}$ and ${\mathcal{G}}$ be Hilbert spaces, and let $A$ be any bounded linear operator from ${\mathcal{H}}$ to ${\mathcal{G}}$. Then ${\mathcal{M}}(A)$ denotes the de Branges-Rovnyak space induced by $A$, that is, ${\mathcal{M}}(A)$ is the Hilbert space consisting of all vectors in $\operatorname{ran}A$ with the pull-back norm

[TABLE]

where $P_{(\ker A)^{\perp}}$ denotes the orthogonal projection onto $(\ker A)^{\perp}$.

The following theorem seems to be well known to specialists in Hilbert space operator theory.

Theorem 4.1**.**

Let $A$ be a bounded linear operator from ${\mathcal{H}}$ to ${\mathcal{G}}$ and let $u$ be a vector in ${\mathcal{G}}$. Then $u$ belongs to $\operatorname{ran}A$ if and only if

[TABLE]

is finite. Further, then $\|u\|_{{\mathcal{M}}(A)}=\gamma$.

The first half of Theorem 4.1 is known as Shmuly’an’s theorem (see Corollary 2 of Theorem 2.1 in Fillmore-Williams [4]). For the second half (norm identity) and also the proof, we referred to Ando [2].

Definition 4.1**.**

Let ${\mathcal{K}}$ be a Kreĭn space. A subspace ${\mathfrak{M}}$ of ${\mathcal{K}}$ is said to be uniformly positive with lower bound $\delta>0$ if

[TABLE]

In particular, for the Kreĭn space defined in Section 3, if ${\mathfrak{M}}$ is uniformly positive with lower bound $\delta>0$, then

[TABLE]

Theorem 4.2**.**

Let ${\mathcal{K}}$ be a Kreĭn space. Every uniformly positive subspace of ${\mathcal{K}}$ with lower bound $\delta>0$ is contained in a maximal uniformly positive subspace with lower bound $\delta>0$, and every maximal uniformly positive subspace is a Hilbert space with the inner product of ${\mathcal{K}}$.

For the details of Definition 4.1 and Theorem 4.2, see Dritschel-Rovnyak [3].

Lemma 4.1**.**

Let $(T_{1},T_{2},T_{3})$ be any triplet in ${\mathfrak{T}}({\mathcal{H}})$. Then, for any $\varepsilon>0$,

[TABLE]

is uniformly positive.

Proof.

For any $\varepsilon>0$ and any vector $x$ in $E_{\varepsilon}^{\perp}{\mathcal{H}}$,

[TABLE]

This concludes the proof. ∎

Let $\widetilde{{\mathfrak{M}}_{\varepsilon}}$ be a maximal uniformly positive subspace containing ${\mathfrak{M}}_{\varepsilon}$. Then, $\widetilde{{\mathfrak{M}}_{\varepsilon}}$ is a Hilbert space with the inner product of ${\mathcal{K}}$ by Theorem 4.2, and hence so is $\overline{{\mathfrak{M}}_{\varepsilon}}$, the closure of ${\mathfrak{M}}_{\varepsilon}$ in $\widetilde{{\mathfrak{M}}_{\varepsilon}}$. We note that $\overline{{\mathfrak{M}}_{\varepsilon}}$ is uniformly positive, that is, there exists some $\delta>0$ such that the following inequality holds:

[TABLE]

Lemma 4.2**.**

Let $(T_{1},T_{2},T_{3})$ be any triplet in ${\mathfrak{T}}({\mathcal{H}})$. Then, for any $\varepsilon>0$, ${\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}}:\overline{{\mathfrak{M}}_{\varepsilon}}\to{\mathcal{H}}$ is bounded as a Hilbert space operator.

Proof.

Since ${\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}}$ is defined everywhere in ${\overline{{\mathfrak{M}}_{\varepsilon}}}$, by the closed graph theorem, it suffices to show that ${\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}}$ is closed. Suppose that

[TABLE]

and

[TABLE]

Then it follows from (4.1) that $x_{n}$, $y_{n}$ and $z_{n}$ converge to $x$, $y$ and $z$ in ${\mathcal{H}}$, respectively. Hence we have that $u=T_{1}x+T_{2}y-T_{3}z$. This concludes the proof. ∎

We will deal with ${\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}}$ as a Hilbert space operator from $\overline{{\mathfrak{M}}_{\varepsilon}}$ to ${\mathcal{H}}$.

Lemma 4.3**.**

Let $(T_{1},T_{2},T_{3})$ be any triplet in ${\mathfrak{T}}({\mathcal{H}})$. Then, for any $\varepsilon>0$, ${\mathcal{M}}(T|_{E_{\varepsilon}^{\perp}{\mathcal{H}}})$ is contractively embedded into ${\mathcal{M}}({\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}})$.

Proof.

Suppose that $u=Tx_{\varepsilon}$ where $x_{\varepsilon}$ is in ${E_{\varepsilon}^{\perp}{\mathcal{H}}}$. In the proof of Theorem 3.1, we showed that

[TABLE]

satisfies ${\mathbb{T}}\mathbf{z}(\varepsilon)=Tx_{\varepsilon}=u$ and

[TABLE]

Moreover, this $\mathbf{z}(\varepsilon)$ belongs to ${\mathfrak{M}}_{\varepsilon}$. These conclude the proof. ∎

The next theorem is a generalization of Corollary 3.1.

Theorem 4.3**.**

Let $(T_{1},T_{2},T_{3})$ be any triplet in ${\mathfrak{T}}({\mathcal{H}})$. Then, for any $\varepsilon>0$,

[TABLE]

as Hilbert spaces.

Proof.

By Lemma 4.3, it suffices to show that ${\mathcal{M}}({\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}})$ is contractively embedded into ${\mathcal{M}}(T|_{E_{\varepsilon}^{\perp}{\mathcal{H}}})$. For any ${\mathbb{T}}\mathbf{x}$ where $\mathbf{x}=(x_{1},x_{2},x_{3})^{t}$ in $\overline{{\mathfrak{M}}_{\varepsilon}}$ and any $x$ in $E_{\varepsilon}^{\perp}{\mathcal{H}}$, we have that

[TABLE]

because $T^{\sharp}\mathbf{x}=(T_{1}^{\ast}x,T_{2}^{\ast}x,T_{3}^{\ast}x)^{t}$ and $(x_{1},x_{2},x_{3})^{t}$ belong to $\overline{{\mathfrak{M}}_{\varepsilon}}$. Hence

[TABLE]

is finite. By Theorem 4.1, ${\mathbb{T}}\mathbf{x}$ belongs to $\operatorname{ran}T|_{E_{\varepsilon}^{\perp}{\mathcal{H}}}$ and

[TABLE]

Thus we have that $\|{\mathbb{T}}\mathbf{x}\|_{{\mathcal{M}}(T|_{E_{\varepsilon}^{\perp}{\mathcal{H}})}}^{2}\leq\|{\mathbb{T}}\mathbf{x}\|_{{\mathcal{M}}({\mathbb{T}}|_{\overline{{\mathfrak{M}}_{\varepsilon}}})}^{2}.$ This concludes the proof. ∎