ACTA issues

On the spectrum of $p$-hyponormal operators

B. P. Duggal

Acta Sci. Math. (Szeged) 63:3-4(1997), 623-637

Abstract. Let ${\cal H}(p)$ (resp. ${\cal H}U(p)$), $0< p< 1/2$, denote the class of $p$-hyponormal operators (resp. $p$-hyponormal operators with equal defect and nullity) on a Hilbert space $H$. Given $A\in{\cal H}(p)$, define $\hat A$ and $\tilde A$ by $\hat A=| A|^{1/2}U| A|^{1/2}$ and $\tilde A=|\hat A|^{1/2}V|\hat A|^{1/2}$, where $U$ and $V$ are as in the polar decompositions $A=U| A|$ and $\hat A=V|\hat A|$. $\tilde A$ is then hyponormal with $\sigma(|\tilde A|)=\sigma(| A|)$ and, if $A\in{\cal H} U(p)$, $\sigma_e(|\tilde A|)=\sigma_e(| A|)$. We use this to prove that ${\cal H}(p)$ operators share a number of spectral properties with hyponormal operators. It is shown that $f(A)$, $A\in{\cal H}U(p)$ and $f$ analytic on $\sigma(A)$, satisfies Weyl's theorem, and $$\| |A|^{2p}-|A^*|^{2p}\|\le\min\big\{\frac p\pi\int_{\sigma(A)}r^{2p-1}dr d\theta, \frac1{\pi^p}\big(\int_{\sigma(A)}rdrd\theta\big)^p\big\}$$ for $A\in{\cal H}(p)$. Also, if an $A\in{\cal H}(p)$ is the sum of a normal and a Hilbert-Schmidt operator, then $A$ is normal.

AMS Subject Classification (1991): 47B20, 47A30

Keyword(s): p, -hyponormal operator, Weyl and essential spectra, area inequality

Received October 11, 1996. (Registered under 5787/2009.)