A Note on Subnormal and Hyponormal Derivations
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In this note we prove that if A and B* are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X − Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X − Xf(B)||₂≤ L||AX − XB||₂, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X 2 L(H) is such that SX − XT belongs to a norm ideal (J, || · ||J ) and prove that f(S)X − Xf(T) 2 J and ||f(S)X − Xf(T)||J ≤ C ||SX − XT||J , for f in a certain class of functions.
Abstract
In this note we prove that if A and B* are subnormal operators and X is a bounded linear operator such that AX − XB is a Hilbert-Schmidt operator, then f(A)X − Xf(B) is also a Hilbert-Schmidt operator and ||f(A)X − Xf(B)||₂≤ L||AX − XB||₂, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X 2 L(H) is such that SX − XT belongs to a norm ideal (J, || · ||J ) and prove that f(S)X − Xf(T) 2 J and ||f(S)X − Xf(T)||J ≤ C ||SX − XT||J , for f in a certain class of functions.
참고문헌 (9)
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