Spectral radius of semi-Hilbertian space operators and its applications

Abstract

In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space H, which are bounded with respect to the seminorm induced by a positive operator A on H. Mainly, we show that r A (T ) ≤ ω A (T ) for every A-bounded operator T , where r A (T ) and ω A (T ) denote respectively the A-spectral radius and the A-numerical radius of T . This allows to establish that r A (T ) = ω 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.

A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications

“…to prove this theorem we only show that ′ = T ij A . Moreover, since A ′ P 2 = O, then by[14, Cor. 2.2] we have ω A ′ (P) =…”

“…In the next proposition, we give some connections between A-numerical radius orthogonality and A-Birkhoff-James orthogonality of operators. Recall from [14] that an operator…”

“…where ω( T ) is the numerical radius of T and ω A (T ) is the A-numerical radius of T , defined as below. [2,3,14]. It is useful to recall that an operator T is called A-self-adjoint if AT is selfadjoint (i.e.…”

Davis–Wielandt shells of semi-Hilbertian space operators and its applications

“…This new concept is intensively studied (see [5,22]). Note that if T ∈ B(H) and satisfies T (N (A)) N (A), then ω A (T ) = +∞ (see [17,Theorem 2.2.]). Moreover, it is easy to see that ω A defines a seminorm on B A 1/2 (H).…”

“…Moreover, it is easy to see that ω A defines a seminorm on B A 1/2 (H). For more information about this concept the reader is invited to consult [17,5,22] and the references therein. One main objective of this paper is to introduce a new type of parallelism for A-bounded operators based on the A-numerical radius and to extend Theorem 1.…”