On properties of class \(A(n)\) and \(n\)-paranormal operators. (English) Zbl 1473.47004

Li, Xiaochun; Gao, Fugen

Abstr. Appl. Anal. 2014, Article ID 629061, 5 p. (2014).

Summary: Let \(n\) be a positive integer, and an operator \(T \in B(\mathcal{H})\) is called a class \(A(n)\) operator if \(\left|T^{1 + n}\right|^{2 / \left(1 + n\right)} \geq | T |^2\) and \(n\)-paranormal operator if \(\|T^{1 + n} x\|^{1 / \left(1 + n\right)} \geq \| T x \|\) for every unit vector \(x \in \mathcal{H}\), which are common generalizations of class \(A\) and paranormal, respectively. In this paper, firstly we consider the tensor products for class \(A(n)\) operators, giving a necessary and sufficient condition for \(T \otimes S\) to be a class \(A(n)\) operator when \(T\) and \(S\) are both non-zero operators; secondly, we consider the properties for \(n\)-paranormal operators, showing that a \(n\)-paranormal contraction is the direct sum of a unitary and a \(C_{. 0}\) completely non-unitary contraction.

MSC:

47B20	Subnormal operators, hyponormal operators, etc.
47A80	Tensor products of linear operators

Keywords:

class \(A(n)\) operator; \(n\)-paranormal operator; tensor product

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