Search

The paper deals with the generalized numerical radius of 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 . Here A is not assumed to be invertible. Mainly, if we denote by ω A (⋅) and ω (⋅) the generalized and the classical numerical radii respectively, we prove that for every A-bounded operator T we have ω A (T) = ω ( A 1 / 2 T (A 1 / 2 ) † ) , where (A 1 / 2 ) † is the Moore-Penrose inverse of A 1 / 2 . In addition, several new inequalities involving ω A (⋅) for single and several operators are established. In particular, by using new techniques, we cover and improve some recent results due to Najafi [Linear Algebra Appl. 2020;588:489–496]. [ABSTRACT FROM AUTHOR]

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree. [ABSTRACT FROM AUTHOR]