ω-Hyponormal operators

The class of -hyponormal operators is introduced. This class contains allp-hyponormal operators. Certain properties of this class of operators are obtained. Among other things, it is shown that ifT is -hyponormal, then its spectral radius and norm are identical, and the nonzero points of its joint point spectrum and point spectrum are identical. Conditions under which a -hyponormal operator becomes normal, self-adjoint and unitary are given.     

w-Hyponormal Operators are Subscalar

We prove that if X is a Banach space, R, S B(X), then RS is subscalar (subdecomposable) if and only if SR is. As corollaries, it is shown that w-hyponormal operators (including p-hyponormal (p > 0) and log-hyponormal operators) and their Aluthge transformations and inverse Aluthge transformations are subscalar.     

PROPERTIES OF p-ω-HYPONORMAL OPERATORS

The approximate point spectrum properties of p-ω-hyponormal operators are given and proved. In faet, it is a generalization of approximate point speetrum properties of ω- hyponormal operators. The relation of spectra and numerical range of p-ω-hyponormal operators is obtained, On the other hand, for p-ω-hyponormal operators T,it is showed that if Y is normal,then T is also normal.     

Onp-hyponormal operators for 0<p<1

The distance formula Tt-I)^–1=[Dist(, (T)]^–1, (T), for hyponormal operators, is generalized top-hyponormal operators for 0<p<1. Several other results involving eigenspaces ofU and |T|, the joint point spectrum, and the spectral radius are also otained, where |T|=(T ^* T)^1/2 andU is the unitary operator in the polar decomposition of thep-hyponormal operatorT=U|T|.     

Isolated point of spectrum ofP-hyponormal, log-hyponormal operators

LetT B(H) be a bounded linear operator on a complex Hilbert spaceH. Let ₀ (T) be an isolated point of (T) and let be the Riesz idempotent for ₀. In this paper, we prove that ifT isp-hyponormal or log-hyponormal, thenE is self-adjoint andE H=ker(H–₀)=ker(H–₀ ^*.This research was supported by Grant-in-Aid Research 1 No. 12640187.     

On log-hyponormal operators

LetTB(H) be a bounded linear operator on a complex Hilbert spaceH.TB(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)log (T ^* T). Since log: (0, )(–,) is operator monotone, for 0<p1, every invertiblep-hyponormal operatorT, i.e., (TT ^*) ^p (T ^* T) ^p , is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform is . Moreover, ifmeas ((T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.This research was supported by Grant-in-Aid Research No. 10640185     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Square of ω-hyponormal operators

In this paper, we show that ifT is a -hyponormal operator, thenT ² is also -hyponormal.     

Structure on Powers of <Emphasis Type="Italic">p</Emphasis>-Hyponormal and log-Hyponormal Operators

Inspired by the problem of powers of hyponormal operators, this paper is to discuss the structure on powers of p-hyponormal and log-hyponormal operators. The structure on powers of operators consists of same-side structure and different-side structure. The same-side structure means relations between and , and the different-side structure means relations between where m, n are positive integers and T is a bounded linear operator on a Hilbert space. Thus, the original problem of powers of hyponormal operators belongs to different-side structure on powers of hyponormal operators. The structure on powers of p-hyponormal operators for p > 0 is emphasized. Also, some applications are obtained.      

A Note on the Spectrum of Invertible p-hyponormal Operators

A bounded linear operator T is clalled p-hyponormal if (T*T)^p ≥ (TT)^p, 0 < p < 1. It is known that for semi-hyponormal operators (p = 1/2), the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. In this paper we prove a somewhat weaker result for invertible p-hyponormal operators for 0 < p < 1/2.