The Putnam-Fuglede property for paranormal and *-paranormal operators

Abstract

Abstract. An operator T ∈ B(H) is said to have the Putnam-Fuglede commutativity property (PF property for short) if T * X = XJ for any X ∈ B(K, H) and any isometry J ∈ B(K) such that T X = XJ * . The main purpose of this paper is to examine if paranormal operators have the PF property. We prove that k * -paranormal operators have the PF property. Furthermore, we give an example of a paranormal without the PF property.

Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-∗-Paranormal Operators

“…These results extend those recently given in [9,13,14] and as applications of our main theorems, we obtain the following:…”

“…The following result was given by Duggal-Kubrusl [22] in the contractive case and by Pagacz [9] in the general case but our proof seems more direct, simpler and gives more explicit decomposition than Pagacz's proof. Proposition 2.…”

“…Many authors extended this theorem for different non-normal classes of operators (see [2,[4][5][6][7][8][9][10][11][12]). In this paper, we shall generalize this theorem to certain (n, k)-quasi- * -paranormal operators.…”

Power boundedm-left invertible operators

“…δ A * ,V (X) = 0) has a non-trivial solution X ∈ B(H), X is also a solution of △ A * ,V (X) = 0 (resp., δ A * ,V (X) = 0). The following theorem is [10, Corollary 2.5] (see also [17]…”

Asymptotic behaviour of Hilbert space operators with applications

“…Theorem 3.2 of [39] tells us that the PF property for a power bounded operator T is equivalent to the condition that T is the orthogonal sum of a unitary and a power bounded operator of class C ·0 . Therefore (i) is an easy consequence of Pagacz's result and Theorem 4.1.…”