An Extension of the Generalized Mean Transform of Hilbert Space Operators

Abstract

In this paper, our objective is to present the generalized ν-mean transform of a Hilbert space operator T, denoted as T_ν(t) with ν, t ∈ [0, 1]. This transform serves as an extension of the generalized mean transform recently introduced by Benhida et al. [Banach J. Math. Anal. 14, 842–855 (2020)], and λ-mean transform studied by Zamani [J. Math. Anal. Appl. 493 (2021)]. Several characterizations involving this new transformation have been established. We have also studied the behaviour of weighted shifts and EP operators under this transform. Among other things, we show that T is an EP operator and R(T_ν(t)) is closed if and only ifT_ν(t) is an EP operator and R(T) = R(T_ν(t)). The relationship between the numerical radius and the operator norm of the generalized ν-mean transform of a Hilbert space operator T, in comparison to those of T itself, is also discussed.