MDPI - Publisher of Open Access Journals

20 pages, 356 KB

Open AccessArticle

New Approach to Generalized Berezin Norms and Rigorous Operator Bounds

by Ghadah Albeladi, Kais Feki and Hala H. Taha

Mathematics 2026, 14(10), 1695; https://doi.org/10.3390/math14101695 - 15 May 2026

Viewed by 204

Let

(H_{Θ}, ⟨ \cdot, \cdot ⟩)

be a reproducing kernel Hilbert space over a non-empty set

Θ

, and let

A

be a non-zero positive operator on

H_{Θ}

. This operator induces a semi-inner product given by [...] Read more.

Let

(H_{Θ}, ⟨ \cdot, \cdot ⟩)

be a reproducing kernel Hilbert space over a non-empty set

Θ

, and let

A

be a non-zero positive operator on

H_{Θ}

. This operator induces a semi-inner product given by

{⟨ ξ, η ⟩}_{A} = ⟨ A ξ, η ⟩

for all

ξ, η \in H_{Θ}

, with the associated seminorm

{∥ ξ ∥}_{A} = \sqrt{{⟨ ξ, ξ ⟩}_{A}}

. The

A

-normalized Berezin number and the

A

-normalized Berezin norm of an

A

-bounded linear operator

C

H_{Θ}

are defined by

b_{A} (C) = \sup_{γ \in Θ_{A}} | {⟨ C {\hat{x}}_{γ}^{A}, {\hat{x}}_{γ}^{A} ⟩}_{A} |

and

{∥ C ∥}_{b_{A}} = \sup_{γ, δ \in Θ_{A}} | {⟨ C {\hat{x}}_{γ}^{A}, {\hat{x}}_{δ}^{A} ⟩}_{A} |

, where

{\hat{x}}_{γ}^{A} = \frac{x_{γ}}{∥ x_{γ} ∥_{A}}

and

Θ_{A} = {γ \in Θ : ∥ x_{γ} ∥_{A} \neq 0}

. The primary aim of this paper is to establish new sharp bounds and inequalities involving these two quantities and related operator-theoretic notions. In doing so, we propose a novel method to address the challenges of operator bounds. Furthermore, we revisit recent results on generalized Berezin norms, in particular those of Huban’s work in 2022. We show that some of these results rely on the incorrect assumption that the

A

-Berezin number coincides with the

A

-Berezin norm for

A

-selfadjoint operators. By providing corrected arguments and employing tools such as the

A

-Cartesian decomposition and the generalized Buzano inequality, we develop a consistent and rigorous framework for the study of generalized Berezin symbols in semi-Hilbertian spaces. Full article

(This article belongs to the Section C: Mathematical Analysis)

12 pages, 253 KB

Open AccessArticle

On the Equality A = A₁A₂ for Linear Relations

by Marcel Roman and Adrian Sandovici

Axioms 2025, 14(4), 239; https://doi.org/10.3390/axioms14040239 - 21 Mar 2025

Cited by 1 | Viewed by 766

Abstract

Assume that A,

A_{1}

, and

A_{2}

are three selfadjoint linear relations (multi-valued linear operators) in a certain complex Hilbert space. In this study, conditions are presented for the multi-valued operator equality

A = A_{1} A_{2}

to hold [...] Read more.

Assume that A,

A_{1}

, and

A_{2}

are three selfadjoint linear relations (multi-valued linear operators) in a certain complex Hilbert space. In this study, conditions are presented for the multi-valued operator equality

A = A_{1} A_{2}

to hold when the inclusion

A \subset A_{1} A_{2}

is assumed to be satisfied. The present study is strongly motivated by the invalidity of a classical result from A. Devinatz, A. E. Nussbaum, and J. von Neumann in the general case of selfadjoint linear relations. Two types of conditions for the aforementioned equality to hold are presented. Firstly, a condition is given in terms of the resolvent sets of the involved objects, which does not depend on the product structure of the right-hand side,

A_{1} A_{2}

. Secondly, a condition is also presented where the structure of the right-hand side is taken into account. This one is based on the notion of the

L

-stability of a linear operator under linear subspaces. It should be mentioned that the classical Devinatz–Nussbaum–von Neumann theorem is obtained as a particular case of one of the main results. Full article

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI