# A Note on Subnormal and Hyponormal Derivations

Lauric, Vasile

• Published : 2008.06.30
• 31 10

#### Abstract

In this note we prove that if A and $B^*$ are subnormal operators and 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 $${\parallel}f(A)X\;-\;Xf(B){\parallel}_2\;\leq\;L{\parallel}AX\;-\;XB{\parallel}_2$$, 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 $X\;{\in}\;\cal{L}(\cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${\parallel}\;{\cdot}\;{\parallel}_J$) and prove that f(S)X - Xf(T) $\in$ J and ${\parallel}f(S)X\;-\;Xf(T){\parallel}_J\;\leq\;C{\parallel}SX\;-\;XT{\parallel}_J$, for f in a certain class of functions.

#### Keywords

subnormal derivations;hyponormal derivations

#### References

1. E. M. Dynkin, Functional calculus based on Cauchy-Green's formula (Russian), Researches in linear operators and function theory. III, Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), (1972), 33-39.
2. E. M. Dynkin, Pseudo-analytic extension of smooth functions. The uniform scale (Russian), Theory of functions and functional analysis, Central Econom. Mat. Inst. Acad. Sci. SSSR, Moscow, (1976), 40-73.
3. T. Furuta, An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality, Proc. Amer. Math. Soc., 81(1981), 240-242. https://doi.org/10.1090/S0002-9939-1981-0593465-4
4. F. Kittaneh, On Lipschitz functions of normal operators, Proc. Amer. Math. Soc., 94(1985), 416-418. https://doi.org/10.1090/S0002-9939-1985-0787884-4
5. M. Putinar and M. Matin, Lectures on Hyponormal Operators, Birkauser Verlag, Boston, 39(1989).
6. D. Voiculescu, Some results on norm ideal perturbation of Hilbert space operators, J. Operator Theory, 2(1979), 3-37.
7. G. Weiss, Fuglede's commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II, J. Operator Theory, 5(1981), 3-16.
8. G. Weiss, The Fuglede commutativity theorem modulo operator ideals, Proc. Amer. math. Soc., 83(1981), 113-118. https://doi.org/10.1090/S0002-9939-1981-0619994-2
9. G. Weiss, An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form MnXNn, Trans. Amer. Math. Soc., 278(1983), 1-20