New results on common properties of bounded linear operators RS and SR

Zeng, Qing Ping; Zhong, Huai Jie

doi:10.1007/s10114-013-1758-3

Cited by 15 publications

(20 citation statements)

References 20 publications

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“…The main result of this note establishes that if

A \in B (X, Y)

and

B, C \in B (Y, X)

satisfy , then

AC - I \in {boldR}_{boldi} ⟷ BA - I \in {boldR}_{boldi}, 1 \leq i \leq 19 .

It not only extends our previous corresponding results in from the special case

B = C

to the general case, but also supplements the results obtained by Corach, Duggal and Harte in from the viewpoint of spectral theory.…”

Section: Introduction and Notationssupporting

confidence: 84%

“…We conclude this note by an example to illustrate that the results obtained in this note are proper generalizations of the corresponding ones in . Example For Banach spaces X and Y , let

T_{1} \in B (Y, X), T_{2} \in B (X, Y)

be arbitrarily given and let

S_{1} \in B (Y, X), S_{2} \in B (X, Y)

be nonzero operators satisfying

S_{1} S_{2} = S_{2} S_{1} = 0

.…”

Section: Resultsmentioning

confidence: 80%

“…The proofs of the following Lemmas – are dependent heavily on the special case

B = C,

which we proved recently in . Lemma If

A \in B (X, Y)

and

B, C \in B (Y, X)

satisfy , then for all

d \in N

R (italicAC - I) + N ({(italicAC - I)}^{d})

is closed if and only if

R (italicBA - I) + N ({(italicBA - I)}^{d})

is closed. Proof Suppose that

R (italicAC - I) + N ({(italicAC - I)}^{d})

is closed.…”

Section: Resultsmentioning

confidence: 99%

“…But at present we are unable to decide whether

AC - I

and

BA - I

share common complementability of kernels. For some other open questions in this direction, we refer the reader to .…”

Section: Introduction and Notationsmentioning

confidence: 99%

“…We now give the definitions of some concrete subsets

{boldR}_{boldi} \subseteq B (X)

1 \leq i \leq 19

. For the fact that R i

(1 \leq i \leq 19)

form a regularity, the reader should refer to . Definition

{boldR}_{bold1} = {T \in scriptB (X) : c (T) = 0}

{boldR}_{bold2} = {T \in scriptB (X) : c (T) < \infty}

{boldR}_{bold3} = {T \in B (X) : there exists d \in N such that c_{d} (T) = 0 and

R (T^{d + 1}) is closed}

{boldR}_{bold4} = {T \in scriptB (X) : c_{n} (T) < \infty for every n \in double-struckN}

{boldR}_{bold5} = {T \in B (X) : there exists d \in N such that c_{d} (T) < \infty and

R (T^{d + 1}) is closed}

{boldR}_{bold6} = {T \in scriptB

…”

Section: Introduction and Notationsmentioning

confidence: 99%

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Common properties of bounded linear operators AC and BA: Spectral theory

Zeng

Zhong

2013

Mathematische Nachrichten

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Let X, Y be Banach spaces, A:X→Y and B, C:Y→X be bounded linear operators satisfying the operator equation ABA=ACA. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common properties of AC−I and BA−I in algebraic viewpoint and also obtained some topological analogues. In this note, we continue to investigate common properties of AC and BA from the viewpoint of spectral theory. In particular, we give an affirmative answer to one question posed by Corach et al. by proving that AC−I has closed range if and only if BA−I has closed range.

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“…The main result of this note establishes that if

A \in B (X, Y)

and

B, C \in B (Y, X)

satisfy , then

AC - I \in {boldR}_{boldi} ⟷ BA - I \in {boldR}_{boldi}, 1 \leq i \leq 19 .

It not only extends our previous corresponding results in from the special case

B = C

to the general case, but also supplements the results obtained by Corach, Duggal and Harte in from the viewpoint of spectral theory.…”

Section: Introduction and Notationssupporting

confidence: 84%

“…We conclude this note by an example to illustrate that the results obtained in this note are proper generalizations of the corresponding ones in . Example For Banach spaces X and Y , let

T_{1} \in B (Y, X), T_{2} \in B (X, Y)

be arbitrarily given and let

S_{1} \in B (Y, X), S_{2} \in B (X, Y)

be nonzero operators satisfying

S_{1} S_{2} = S_{2} S_{1} = 0

.…”

Section: Resultsmentioning

confidence: 80%

“…The proofs of the following Lemmas – are dependent heavily on the special case

B = C,

which we proved recently in . Lemma If

A \in B (X, Y)

and

B, C \in B (Y, X)

satisfy , then for all

d \in N

R (italicAC - I) + N ({(italicAC - I)}^{d})

is closed if and only if

R (italicBA - I) + N ({(italicBA - I)}^{d})

is closed. Proof Suppose that

R (italicAC - I) + N ({(italicAC - I)}^{d})

is closed.…”

Section: Resultsmentioning

confidence: 99%

“…But at present we are unable to decide whether

AC - I

and

BA - I

share common complementability of kernels. For some other open questions in this direction, we refer the reader to .…”

Section: Introduction and Notationsmentioning

confidence: 99%

“…We now give the definitions of some concrete subsets

{boldR}_{boldi} \subseteq B (X)

1 \leq i \leq 19

. For the fact that R i

(1 \leq i \leq 19)

form a regularity, the reader should refer to . Definition

{boldR}_{bold1} = {T \in scriptB (X) : c (T) = 0}

{boldR}_{bold2} = {T \in scriptB (X) : c (T) < \infty}

{boldR}_{bold3} = {T \in B (X) : there exists d \in N such that c_{d} (T) = 0 and

R (T^{d + 1}) is closed}

{boldR}_{bold4} = {T \in scriptB (X) : c_{n} (T) < \infty for every n \in double-struckN}

{boldR}_{bold5} = {T \in B (X) : there exists d \in N such that c_{d} (T) < \infty and

R (T^{d + 1}) is closed}

{boldR}_{bold6} = {T \in scriptB

…”

Section: Introduction and Notationsmentioning

confidence: 99%

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Common properties of bounded linear operators AC and BA: Spectral theory

Zeng

Zhong

2013

Mathematische Nachrichten

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New results on common properties of the products AC and BA, II

Zeng

Yan

Zhang

2020

Mathematische Nachrichten

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In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If are elements in a unital associative ring satisfying , then von Neumann regularity (resp. generalized Fredholmness relative to an ideal of ) of is converted into that of . (ii) If are bounded linear operators satisfying , then and share common complementability of kernels and ranges. (iii) If are elements in a unital semisimple Banach algebra satisfying and is a trace ideal of such that soc kh(soc, then and share common Fredholmness relative to and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra.

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Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting

Arzini,

Jaatit

2024

Adv. Appl. Clifford Algebras

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New results on common properties of bounded linear operators RS and SR

Cited by 15 publications

References 20 publications

Common properties of bounded linear operators AC and BA: Spectral theory

Common properties of bounded linear operators AC and BA: Spectral theory

New results on common properties of the products AC and BA, II

Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting

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