Let $T$ be a bounded linear operator on a Hilbert space $mathscr{H}$. We say that $T$ has the hyponormal property if there exists a function $f$, continuous on an appropriate set so that $f(|T|)geq f(|T^ast|)$. We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function $f$ we introduce the $f$-Aluthge tran...

In this paper, we initiate the notion of k^{th} order slant Hankel operators on L^2(T^n) for k greater than or equal to 2 and n 1 where T^n denotes n-torus. We give necessary sufficient condition a bounded operator be discuss their commutative, compactness, hyponormal isometric property.

We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal.

Let T=H+iK be hyponormal and Q be a strictly monotone increasing continuous function on s(H ). We define ~ Q(T ) by ~ Q(T )=Q(H )+iK. In this paper, we show that if z is an isolated eigenvalue of ~ Q(T ), then the corresponding Riesz projection is self-adjoint. Also we introduce Xia spectrum and study the existence of an invariant subspace of an operator ~ Q(T ).