An Operator Inequality Which Implies Paranormality

Abstract

Abstract. Let T be a bounded linear operator on a Hilbert space. Among other things, it is shownand |T * 2 | |T * | 2 , and (3) if T and T * are w -hyponormal, and either ker T ⊆ ker T * or ker T * ⊆ ker T , then T is normal.Mathematics subject classification (1991): 47B20, 47A63. Key words and phrases: Hyponormal operator, p -hyponormal operator, log-hyponormal operator, w -hyponormal operator, paranormal operator.

TENSOR PRODUCTS OF LOG-HYPONORMAL AND OF CLASS $A(s,t)$ OPERATORS

“…A class A(1, 1) operator is called a class A operator and a class wA(1/2, 1/2) operator is called a w-hyponormal operator ( [2,8]). T is said to be paranormal if Tx 2 ≤ T 2 x x for x ∈ H. It is known that class A operators are paranormal.…”

Mosaic and Principal Functions of log-hyponormal Operators

“…Hence, Lemma 2 holds for invertible whyponormal operators. By Theorem 2 of [2], it holds that if T is log-hyponormal, then T is w-hyponormal. Therefore we have ker(T 1 ) = ker(T * 1 ).…”

“…So we omit the proof. (2) | are the polar decompositions on E and E ⊥ respectively, then T = U (1) ⊕ U (2) · |T (1) | ⊕ |T (2) | is the polar decomposition of T. Lemma 8. Let T = U |T | ∈ SHU and E be the maximal reducing subspace such that T |E is normal.…”

On Quasisimilarity for Log-Hyponormal Operators

“…An operator X ∈ L(H 2 , H 1 ) is called a quasiaffinity if X is injective and has dense range R(X). For T 1 ∈ L(H 1 ) and T 2 ∈ L(H 2 ), if there exist quasiaffinities X ∈ L(H 2 , H 1 ) and 2 and YT 1 = T 2 Y , then we say that T 1 and T 2 are quasisimilar.…”

“…The idea of a log-hyponormal operator is due to Ando [3] and the first paper in which log-hyponormality appeared is [9]. See [2,16,17, 18] for properties of log-hyponormal operators.…”