We extend a result concerning λ-commuting normal operators with empty point spectrum. More precisely, we prove that for a hyponormal operator T with empty point spectrum for which there exists a Hilbert-Schmidt operator K such that TK = λKT + μK for some |λ| < 1 and μ ∈ C, implies K = 0.

We consider integration and double integration operators, the Hardy operator, and multiplication and composition operators on Lebesgue space $L_{p}\left[ 0,1\right] $ and Sobolev spaces $W_{p}^{\left( n\right) }\left[ 0,1\right] $ and $W_{p}^{\left( n\right) }\left( \left[ 0,1\right] \times\left[ 0,1\right] \right) ,$ and we study their properties. In particular, we calculate norm and spectral multiplicity of the Hardy operator and some multiplication operators, investigate its extended eigenvectors, characterize some composition operators in terms of the extended eigenvectors of the Hardy operator, and calculate the numerical radius of the integration operator on the real $L_{2}\left[ 0,1\right] $ space. The main method for our investigation is the so-called Duhamel products method. Some other questions are also discussed and posed.

background

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A remark on λ-interwining hyponormal operators

Some concrete operators and their properties

We prove that if A and B* are subnormal operators acting on a Hilbert space, then for every bounded linear operator X, the Hilbert-Schmidt norm of AX XB is greater than or equal to the Hilbert-Schmidt norm of A*X XB*. In particular, AX = XB implies A*X = XB*. In addition, if we assume X is a Hilbert-Schmidt operator, we can relax the subnormality conditions to hyponormality and still retain the inequality. 1. In this paper an operator means a bounded linear operator on a separable infinite dimensional Hilbert space H. Let B(H) and C2 denote the class of all bounded linear operators acting on H and the Hilbert-Schmidt class in B(H), respectively. It is known that C2 forms a two-sided ideal in the algebra B(H) and C2 is itself a Hilbert space for the inner product (X, Y) = 2(Xej, Yej) = Tr(Y*X) =Tr(XY*) where { ej} is any orthonormal basis of H and Tr() denotes the trace. In what follows, 11 112 denotes the Hilbert-Schmidt norm. An operator T is called subnormal if T has a normal extension and hyponormal if T* T > TT*. The inclusion relation of these classes of nonnormal operators is as follows: Normal ; Subnormal i Hyponormal. The above inclusions are all proper [5, Problem 160, p. 101]. THEOREM A [9]. If A and B are normal, then IlAX XBI2 = IIA*X XB*112 for every X E B(H). THEOREM B [3]. If A and B* are subnormal operators and if X is an operator such that AX = XB, then A*X = XB*. In this paper we integrate Theorem A and Theorem B in order to prove a slightly stronger Theorem 1. Moreover in our Theorem 2 we have an extension of Weiss [8, Theorem 3] and Berberian [2, Theorem]. Finally we shall pose an open problem with respect to Theorem 1. Received by the editors July 3, 1979 and, in revised form, October 25, 1979 and February 15, 1980. 1980 Mathematics Subject Classification. Primary 47B10, 47B15, 47B20; Secondary 47A30.

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A remark on λ-interwining hyponormal operators

Citations

Some concrete operators and their properties

References

An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality

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