A Note on Subnormal and Hyponormal Derivations
نویسنده
چکیده
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 ∈ L(H) is such that SX −XT belongs to a norm ideal (J, || · ||J) and prove that f(S)X −Xf(T ) ∈ J and ||f(S)X −Xf(T )||J ≤ C ||SX −XT ||J , for f in a certain class of functions.
منابع مشابه
Some Estimates of Certain Subnormal and Hyponormal Derivations
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 belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and X ∈ L H is such that SX − XT belo...
متن کاملHyponormal Operators with Rank-two Self-commutators
In this paper it is shown that if T ∈ L(H) satisfies (i) T is a pure hyponormal operator; (ii) [T ∗, T ] is of rank-two; and (iii) ker [T ∗, T ] is invariant for T , then T is either a subnormal operator or the Putinar’s matricial model of rank two. More precisely, if T |ker [T∗,T ] has the rank-one self-commutator then T is subnormal and if instead T |ker [T∗,T ] has the ranktwo self-commutato...
متن کاملJointly Hyponormal Pairs of Commuting Subnormal Operators Need Not Be Jointly Subnormal
We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
متن کاملPropagation Phenomena for Hyponormal 2-variable Weighted Shifts
We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for 1-variable weighted shifts. In 1variable, it is w...
متن کاملExistence of Non-subnormal Polynomially Hyponormal Operators
In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality impl...
متن کامل