On 𝑝-hyponormal contractions

Duggal, B. P.

doi:10.1090/S0002-9939-1995-1264808-3

On $p$-hyponormal contractions
HTML articles powered by AMS MathViewer

by B. P. Duggal PDF

Proc. Amer. Math. Soc. 123 (1995), 81-86 Request permission

Abstract:

The contraction A on a Hilbert space H is said to be p-hyponormal, $0 < p < 1$, if ${({A^ \ast }A)^p} \geq {(A{A^ \ast })^p}$. Let A be an invertible p-hyponormal contraction. It is shown that A has ${C_{.0}}$ completely nonunitary part. Now let H be separable. If A is pure and the defect operator ${D_A} = {(1 - {A^ \ast }A)^{1/2}}$ is of Hilbert-Schmidt class, then $A \in {C_{10}}$. Let ${B^ \ast }$ be a contraction such that ${B^\ast }$ has ${C_{.0}}$ completely nonunitary part, ${D_{{B^ \ast }}}$ is of Hilbert-Schmidt class, and ${B^ \ast }$ satisfies the property that if the restriction of ${B^ \ast }$ to an invariant subspace is normal, then the subspace reduces ${B^ \ast }$. It is shown that if $AX = XB$ for some quasi-affinity X, then A and B are unitarily equivalent normal contractions.

References

Ariyadasa Aluthge, On $p$-hyponormal operators for $0<p<1$, Integral Equations Operator Theory 13 (1990), no. 3, 307–315. MR 1047771, DOI 10.1007/BF01199886

Properties of p-hyponormal operators

B. P. Duggal, Contractions with a unitary part, J. London Math. Soc. (2) 31 (1985), no. 1, 131–136. MR 810570, DOI 10.1112/jlms/s2-31.1.131
B. P. Duggal, On intertwining operators, Monatsh. Math. 106 (1988), no. 2, 139–148. MR 968331, DOI 10.1007/BF01298834
B. P. Duggal, A generalized commutativity theorem, Z. Anal. Anwendungen 10 (1991), no. 3, 265–274. MR 1155610, DOI 10.4171/ZAA/451
Takayuki Furuta, $A\geq B\geq 0$ assures $(B^rA^pB^r)^{1/q}\geq B^{(p+2r)/q}$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$, Proc. Amer. Math. Soc. 101 (1987), no. 1, 85–88. MR 897075, DOI 10.1090/S0002-9939-1987-0897075-6
Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Béla Sz.-Nagy and Ciprian Foiaş, On injections, intertwining operators of class $C_{0}$, Acta Sci. Math. (Szeged) 40 (1978), no. 1-2, 163–167. MR 487520
Joseph G. Stampfli and Bhushan L. Wadhwa, An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana Univ. Math. J. 25 (1976), no. 4, 359–365. MR 410448, DOI 10.1512/iumj.1976.25.25031
Katsutoshi Takahashi and Mitsuru Uchiyama, Every $C_{00}$ contraction with Hilbert-Schmidt defect operator is of class $C_{0}$, J. Operator Theory 10 (1983), no. 2, 331–335. MR 728912

On the non-normal operators—semihyponormal operators

23

Daoxing Xia, Spectral theory of hyponormal operators, Operator Theory: Advances and Applications, vol. 10, Birkhäuser Verlag, Basel, 1983. MR 806959, DOI 10.1007/978-3-0348-5435-1