Spectral radius of semi-Hilbertian space operators and its applications

Kais Feki

RECORD

Spectral radius of semi-Hilbertian space operators and its applications

Kais Feki

Functional Analysis · Submitted 11 Nov 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Mainly, we show that $r_A(T)\leq \omega_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $\omega_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,863 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.87 · compares with prior results 0.10 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq \omega_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $\omega_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=\omega_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover, some characterizations of $A$-normaloid and $A$-spectraloid operators are given.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$.	contribution 0.98	0.47	0.04	asserted 0.64
s1	Mainly, we show that $r_A(T)\leq \omega_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $\omega_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$.	result 0.99	0.94	0.89	asserted 1.00
s2	This allows to establish that $r_A(T)=\omega_A(T)=\\|T\\|_A$ for every $A$-normaloid operator $T$, where $\\|T\\|_A$ is denoted to be the $A$-operator seminorm of $T$.	result 0.98	0.90	0.21	asserted 1.00
s3	Moreover, some characterizations of $A$-normaloid and $A$-spectraloid operators are given.	contribution 0.63	0.90	0.05	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq \omega_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $\omega_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=\omega_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover, some characterizations of $A$-normaloid and $A$-spectraloid operators are given.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

3 arXiv papers cite this in 16 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 11 sentences without a confident label

Cites it as background 2 papers

Recently, K.Feki introduced in [ref] the concept of $A$-normaloid operators as follows.

Davis-Wielandt shells of semi-Hilbertian space operators and its applications · The $A$-Davis-Wielandt shell of operators · 0.89
Recall from [ref] that an operator $T∈ℬ_A^1/2(ℋ)$ is said to be $A$-normaloid if $r_A(T)=T_A$, where [equation] Moreover, it was shown in [ref] that an operator $T∈ℬ_A^1/2(ℋ)$ is $A$-normaloid if and only if $ω_A(T)=T_A$.

$A$-Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications · $A$-numerical radius orthogonality and parallelism · 0.89

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.