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Let \mathcal {B}(H) be the algebra of all bounded operators on a Hilbert space H. Let T=V|T| be the polar decomposition of an operator T\in \mathcal {B}(H). The mean transform of T is defined by M(T)=\frac {T+|T|V}{2}. In this paper, we discuss several properties related to the spectrum, the kernel, the image, and the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators such as positive, normal, unitary, hyponormal, and co-hyponormal operators. [ABSTRACT FROM AUTHOR]

Let H be a complex, separable Hilbert space and let B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm |⋅|u on B(H) and two linear operators A and B in B(H), we shall say that A and B are polynomially isometric relative to |⋅|u if |p(A)|u = |p(B)|u for all polynomials p. In this paper, we examine to what extent an operator A being polynomially isometric to a normal operator N implies that A is itself normal. More explicitly, we first show that if |⋅|u is any unitarily-invariant norm on Mn(C), if A, N ∈ Mn(C) are polynomially isometric and N is normal, then A is normal. We then extend this result to the infinite-dimensional setting by showing that if A, N ∈ B(H) are polynomially isometric relative to the operator norm and N is a normal operator whose spectrum neither disconnects the plane nor has interior, then A is normal, while if the spectrum of N is not of this form, then there always exists a nonnormal operator B such that B and N are polynomially isometric. Finally, we show that if A and N are compact operators with N normal, and if A and N are polynomially isometric with respect to the (c,p)-norm studied by Chan, Li, and Tu, then A is again normal. [ABSTRACT FROM AUTHOR]

An operator T on a complex Hilbert space H is said to be skew symmetric if there exists a conjugate-linear, isometric involution C : H → H so that CTC = --T*. In this paper, we shall give two structure theorems for skew symmetric normal operators. [ABSTRACT FROM AUTHOR]

In this paper we investigate multinormed W*-algebras in terms of the central topologies of W*-algebras. The main result asserts that each multinormed W*-algebra can be realized as a local von Neumann algebra on a certain domain in a Hilbert space. Moreover, it admits the predual (unique up to an isometry), which is the ℓ1-normed space. In the normed case the assertion is reduced to the known Sakai theorem. [ABSTRACT FROM AUTHOR]