The generalized Ces\`{a}ro matrices of order three are supraposinormal on $\ell^2$

TL;DR

This paper extends the analysis of generalized Cesàro matrices to order three, demonstrating they are supraposinormal on , and explores their related operator properties, suggesting further research on higher orders.

Contribution

The paper adapts previous methods to prove that generalized Cesàro matrices of order three are supraposinormal on , providing new insights into their operator properties.

Findings

Order three matrices are supraposinormal on 

Results imply posinormality, coposinormality, hyponormality

Conjecture proposed for higher orders

Abstract

The procedure that was used in an earlier paper on the generalized Ces\`{a}ro matrices of order two is adapted here to show that generalized Ces\`{a}ro matrices of order three are supraposinormal on $\ell^2$. This leads to information about their posinormality, coposinormality, and hyponormality. The reader is then invited to formulate a conjecture regarding generalized Ces\`{a}ro matrices of other orders.

Equations

$$AQ{A}^{\ast }=A^{\ast }PA,$$

$$<(A^{\ast }A-A{A}^{\ast })f,f>\ge 0$$

$$Q\ge I\ge P\ge 0,$$

$$⟨(A^{\ast }A-A{A}^{\ast })f,f⟩=⟨(A^{\ast }A-A^{\ast }PA+AQ{A}^{\ast }-A{A}^{\ast })f,f⟩$$

$$=⟨(I-P)Af,Af⟩+⟨(Q-I)A^{\ast }f,A^{\ast }f⟩\ge 0$$

$$(C^{(\alpha )},\beta {)}_{ij}=\frac{\Gamma (i+\alpha +1)}{\Gamma (i-j+1)}\cdot \frac{\beta \Gamma (i-j+\beta )}{\Gamma (i+\alpha +\beta +1)}.$$

$$m_{ij}=(C^{(\alpha )},1{)}_{ij}=\{\begin{array}{ll}\frac{1}{i+1+\alpha }& \text{ for }0\le j\le i\\ 0& \text{ for }j>i.\end{array}$$

$$m_{ij}=\{\begin{array}{ll}\frac{2(i+1-j)}{(i+1+\alpha )(i+2+\alpha )}& \text{ for }0\le j\le i\\ 0& \text{ for }j>i,\end{array}$$

$$m_{ij}=\{\begin{array}{ll}\frac{3(i+1-j)(i+2-j)}{(i+1+\alpha )(i+2+\alpha )(i+3+\alpha )}& \text{ for }0\le j\le i\\ 0& \text{ for }j>i.\end{array}$$

$$P:\equiv\operatorname{diag}\Big{\{}\frac{(n+1+\alpha)(n+2+\alpha)(n+3+\alpha)}{(n+4+\alpha)(n+5+\alpha)(n+6+\alpha)}:n\geq 0\Big{\}},$$

$$\sum _{k=0}^{\infty }\frac{9(j+1-i+k)(j+2-i+k)(k+1)(k+2)}{(j+1+k+\alpha )(j+2+k+\alpha )(j+3+k+\alpha )(j+4+k+\alpha )(j+5+k+\alpha )(j+6+k+\alpha )}.$$

$$s(k)-s(k+1)$$

$$s(k):\equiv \frac{a{k}^4+b{k}^3+c{k}^2+dk+e}{(j+1+k+\alpha )(j+2+k+\alpha )(j+3+k+\alpha )(j+4+k+\alpha )(j+5+k+\alpha )}$$

$$a=9,b=9(8+2\alpha -i+3j),c=3(71-15i+i^2+57j-5ij+10j^2+42\alpha +6{\alpha }^2-3\alpha i+15\alpha j),$$

$$d=\frac{3}{2}(180-48i+6i^2+236j-36ij+i^{2}j+90j^2-5i{j}^2+10j^3+188\alpha +60{\alpha }^2+6{\alpha }^3-24\alpha i-3{\alpha }^{2}i)$$

$$+\frac{3}{2}(\alpha i^2+144\alpha j+21{\alpha }^{2}j-8\alpha ij+25\alpha j^2),$$

$$e=\frac{3}{10}(4+j+\alpha )(5+j+\alpha )(20+24\alpha +6{\alpha }^2-6i-3\alpha i+i^2+30j+15\alpha j-5ij+10j^2).$$

$$s(0)=\frac{3(20+24\alpha +6{\alpha }^2-6i-3\alpha i+i^2+30j+15\alpha j-5ij+10j^2)}{10(j+1+\alpha )(j+2+\alpha )(j+3+\alpha )}.$$

$$Q=\left(\begin{array}[]{ccccccc}q_{00}&q_{01}&q_{02}&0&0&0&\ldots\\ q_{10}&q_{11}&q_{12}&0&0&0&\ldots\\ q_{20}&q_{21}&q_{22}&0&0&0&\ldots\\ 0&0&0&1&0&0&\ldots\\ 0&0&0&0&1&0&\ldots\\ 0&0&0&0&0&1&\ldots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right);$$

$$x_{ij}=\{\begin{array}{ll}\frac{(10j^2+(15\alpha +30)j+20+24\alpha +6{\alpha }^2)(1+\alpha )(2+\alpha )(3+\alpha )}{20(j+1+\alpha )(j+2+\alpha )(j+\alpha +3)}& \text{ for }i=0\\ \frac{((10-20\alpha )j^2+(10-50\alpha -30{\alpha }^2)j-33\alpha -45{\alpha }^2-12{\alpha }^3)(2+\alpha )(3+\alpha )}{20(j+1+\alpha )(j+2+\alpha )(j+\alpha +3)}& \text{ for }i=1\\ \frac{((20+10{\alpha }^2)j^2+(-20+30\alpha +35{\alpha }^2+15{\alpha }^3)j+18\alpha +39{\alpha }^2+27{\alpha }^3+6{\alpha }^4)(3+\alpha )}{20(j+1+\alpha )(j+2+\alpha )(j+\alpha +3)}& \text{ for }i=2\\ \frac{3(j+1-i)(j+2-i)}{(j+1+\alpha )(j+2+\alpha )(j+\alpha +3)}& \text{ for }j\ge i\ge 3\\ 0& \text{ for }i\ge 3,j<i.\end{array}$$

$$\frac{3(i+1)(i+2)}{(i+1+\alpha )(i+2+\alpha )(i+3+\alpha )}\cdot x_{0j}$$

$$+\frac{3i(i+1)}{(i+1+\alpha )(i+2+\alpha )(i+3+\alpha )}\cdot x_{1j}$$

$$+\frac{3(i-1)i}{(i+1+\alpha )(i+2+\alpha )(i+3+\alpha )}\cdot x_{2j}$$

$$+\sum _{k=2}^{i-1}\frac{3(i-k)(i-k+1)}{(i+1+\alpha )(i+2+\alpha )(i+3+\alpha )}\cdot \frac{3(j-k)(j-k+1)}{(j+1+\alpha )(j+2+\alpha )(j+\alpha +3)}$$

$$=\frac{3(20+24\alpha +6{\alpha }^2-6i-3\alpha i+i^2+30j+15\alpha j-5ij+10j^2)}{10(j+1+\alpha )(j+2+\alpha )(j+3+\alpha )},$$

$$\det\left(\begin{array}[]{cc}q_{00}&q_{01}\\ q_{10}&q_{11}\end{array}\right)=\frac{(1+\alpha)(2+\alpha)^{2}(3+\alpha)^{2}(4+\alpha)(20+15\alpha+3\alpha^{2})}{2880}$$

$$\det\left(\begin{array}[]{ccc}q_{00}&q_{01}&q_{02}\\ q_{10}&q_{11}&q_{12}\\ q_{20}&q_{21}&q_{22}\end{array}\right)=\frac{(1+\alpha)(2+\alpha)^{2}(3+\alpha)^{3}(4+\alpha)^{2}(5+\alpha)}{8640},$$

$$\det\left(\begin{array}[]{cc}q_{00}-1&q_{01}\\ q_{10}&q_{11}-1\end{array}\right)=\frac{(\alpha-1)\alpha^{2}(1+\alpha)(2308+1860\alpha+521\alpha^{2}+60\alpha^{3}+3\alpha^{4})}{2880}$$

$$\det\left(\begin{array}[]{ccc}q_{00}-1&q_{01}&q_{02}\\ q_{10}&q_{11}-1&q_{12}\\ q_{20}&q_{21}&q_{22}-1\end{array}\right)=\frac{(\alpha-2)(\alpha-1)^{2}\alpha^{3}(1+\alpha)^{2}(2+\alpha)}{8640},$$

Full text

1. Introduction

The concept of supraposinormality for operators on a Hilbert space $H$ was introduced and investigated in [2]. The operator $A\in\mathcal{B}(H)$, the set of bounded linear operators on $H$, is supraposinormal if there exist positive operators $P$ and $Q\in\mathcal{B}(H)$ such that

[TABLE]

where at least one of $P$, $Q$ has dense range. The ordered pair $(Q,P)$ is referred to as an interrupter pair associated with $A$. A posinormal operator is a particular case of a supraposinormal with $Q=I$, and a coposinormal operator is a particular case of a supraposinormal with $P=I$. From [2, Theorem $1$] we know that if $P$ is invertible, then the supraposinormal operator $A$ is coposinormal; and we know that if $Q$ is invertible, then the supraposinormal operator $A$ is posinormal.

Recall that the operator $A\in\mathcal{B}(H)$ is hyponormal if

[TABLE]

for all $f\in H$. The following theorem will be of use in the next section.

Theorem 1.1**.**

If $A$ is supraposinormal operator on $H$ with $AQA^{*}=A^{*}PA$ and

[TABLE]

then $A$ is hyponormal.

Proof.

Assume the hypothesis is satisfied. Then

[TABLE]

[TABLE]

for all f in $H$, so $A$ is hyponormal. ∎

As was noted in [3], the generalized Cesàro matrices of order $\beta$, $(C^{(\alpha)},\beta)$, for $\alpha,\beta\geq 0$ are the lower triangular infinite matrices given by

[TABLE]

For fixed $\alpha\geq 0$ and $\beta=1$, one obtains the generalized Cesàro matrices of order one, $(C^{(\alpha)},1)$, with entries

[TABLE]

for all $i$. We note that $(C^{(\alpha)},1)\in\mathcal{B}(\ell^{2})$ for all $\alpha>-1$, not just for $\alpha\geq 0$. The operators $(C^{(\alpha)},1)$ are known to be supraposinormal, posinormal, and coposinormal for all $\alpha>-1$, and $(C^{(\alpha)},1)$ is hyponormal for all $\alpha\geq 0$.

Using a diagonal operator $P$ and a “nearly” diagonal operator $Q$, it was shown in [3] that the matrix $M:\equiv(C^{(\alpha)},2)$, the generalized Cesàro matrices of order two, with entries given by

[TABLE]

are supraposinormal, posinormal, and coposinormal for $\alpha>-1$ and hyponormal for $\alpha\in{0}\cup[1,\infty)$.

2. Main Results

Under consideration here will be the matrix $M:\equiv(C^{(\alpha)},3)$ whose entries are as follows:

[TABLE]

We note that $M\in\mathcal{B}(\ell^{2})$ for all $\alpha>-1$. The procedure followed below is similar to that used in [3], and some of the calculations have been aided by the Sagemath software system [4].

Theorem 2.1**.**

The operator $M\in\mathcal{B}(\ell^{2})$ is supraposinormal for $\alpha>-1$.

Proof.

We take

[TABLE]

and then compute $M^{*}PM$.

For $j\geq i$, the ($i$, $j$)th entry of $M^{*}PM$ is

[TABLE]

The series is telescoping, as can be seen by rewriting the summand as

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Consequently, for $j\geq i$, the ($i$, $j$)th entry of $M^{*}PM$ in simplified form is

[TABLE]

The expression for $i>j$ is similar, with the roles of $i$ and $j$ reversed.

Suppose that $Q=[q_{ij}]$ is given by

[TABLE]

where $q_{00}=\frac{(10+12\alpha+3\alpha^{2})(1+\alpha)(2+\alpha)(3+\alpha)}{60}$, $q_{10}=q_{01}=-\frac{\alpha(11+4\alpha)(1+\alpha)(2+\alpha)(3+\alpha)}{40}$,

$q_{20}=q_{02}=\frac{\alpha(3+2\alpha)(1+\alpha)(2+\alpha)(3+\alpha)}{40}$, $q_{11}=\frac{(5-\alpha+15\alpha^{2}+6\alpha^{3})(2+\alpha)(3+\alpha)}{30}$,

$q_{12}=q_{21}=-\frac{\alpha(1+\alpha)(2+\alpha)(3+\alpha)(1+4\alpha)}{40}$, $q_{22}=\frac{(20-6\alpha+7\alpha^{2}+6\alpha^{3}+3\alpha^{4})(3+\alpha)}{60}$,

$q_{ii}=1$ for all $i=j\geq 3$, and $q_{ij}=0$ otherwise.

If $X:\equiv QM^{*}$, then $X=[x_{ij}]$ has entries

[TABLE]

For $j\geq i\geq 3$, the ($i$,$j$)th entry of $MQM^{*}=MX$ is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as is best verified with the aid of computational software. The cases $j\geq i=0$, $j\geq i=1$, and $j\geq i=2$ are left to the reader. The computations for $i\geq j$ are similar, so it follows that $M^{*}PM=MQM^{*}$. It is clear that $P$ has dense range, so $M$ is supraposinormal. ∎

Corollary 2.2**.**

The operator $M$ is both posinormal and coposinormal for $\alpha>-1$.

Proof.

Since

[TABLE]

and

[TABLE]

it is clear that $Q$ is positive and invertible for $\alpha>-1$. Since $P$ is also clearly positive and invertible for $\alpha>-1$, it follows from [2, Theorem $1$] that $M$ is both posinormal and coposinormal when $\alpha>-1$. ∎

Corollary 2.3**.**

If $n$ is a positive integer and $\alpha>-1$, then $M^{n}$ is both posinormal and coposinormal.

Proof.

This follows from Corollary $2.2$ and [1, Corollary $1$ (b)]. ∎

Corollary 2.4**.**

The operator $M$ is hyponormal for $\alpha\in\{0,1\}\cup[2,\infty)$.

Proof.

Since

[TABLE]

and

[TABLE]

are nonnegative for $\alpha\in\{0,1\}\cup[2,\infty)$, it follows from Theorem $1.1$ that $M$ is hyponormal for those values of $\alpha$. ∎

3. Closing Remark

On the basis of what is now known regarding supraposinormality, posinormality, coposinormality, and hyponormality for the generalized Cesàro matrices of orders one, two, and three, the reader is invited to formulate a conjecture concerning these properties for the generalized Cesàro matrices of order $k$ when $k\in\mathbb{Z}+$.