# On a question of N. Salinas

01 Jan 1986-Vol. 98, Iss: 1, pp 94-96

Abstract: In [5], Salinas asked the following question: If T = (Ti,.. , Tn) consists of commuting hyponormal operators, is it true that (1) 5(T A) = d(A, a, (T)) and (2) r, (T) = I IDT II? He proved that, for a doubly commuting n-tuple of quasinormal operators, (2) was true and (1) was true for A = 0. In this paper we shall show that (2) holds for a doubly commuting n-tuple of hyponormal operators and give an example of a subnormal operator which does not satisfy (1).

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Abstract: We introduce and discuss a class of operator tuples, which we call completely hyperexpansive tuples of finite order This class is in some sense antithetical to the class of completely hypercontractive tuples of finite order studied in the prequel of this paper Motivated by Shimorin's notion of Cauchy dual operator, we also discuss a transform which sends certain completely hyperexpansive multishifts of finite order k to completely hypercontractive multishifts of respective order

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Abstract: Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible «-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate. 1. Introduction. 1. Let T be a (bounded linear) operator on a Banach space %. T is said to be invertible if there exists an operator S on % such that TS = ST = 1%, the identity operator on 9C. By the Open Mapping Theorem, this is equivalent to ker T = (0) and R(T) = range of T = %. The last formulation does not rely upon the existence of an inverse for T, but rather on the action of the operator T. When T is replaced by an «-tuple of commuting operators, several definitions of nonsingular- ity exist. J. L. Taylor (19) has obtained one which reflects the actions of the operators, by considering the Koszul complex associated with the «-tuple. 2. In this paper we develop a general "several variables" theory on the basis of Taylor's work and study commuting and almost commuting (= commuting mod- ulo the compacts) «-tuples of operators on a Hilbert space %. We obtain a characterization of joint invertibility in terms of the invertibility of a single operator, which is essential for our approach. From that we get a number of corollaries which generalize nicely the known elementary results in "one variable". At the same time, the referred characterization allows us to define a continuous, invariant under compact perturbations, integer-valued index on the class of Fred- holm «-tuples (those almost commuting «-tuples which are invertible in the Calkin algebra). This index extends the classical one for Fredholm operators. We prove that an almost commuting «-tuple of essentially normal operators with all commu- tators in trace class has index zero (« > 2) and that a natural generalization of Atkinson's theorem holds for «-tuples.

118 citations

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Abstract: The joint numerical status of commuting bounded operators Ai and A2 on a Hubert space is defined as {{φiA^y φ(A2)) such that φ is a state on the C*-algebra generated by Ax and A2}. It is shown that if At and A2 are commuting normal operators then their joint numerical status equals the closure of their joint numerical range. It is also shown that certain points in the boundary of the joint numerical range are joint approximate reducing eigenvalues. The joint numerical range of Ax and A2 denoted by w(Alf A2) is jX, x), (A2x, x)) such that xeH and ||g|| = 1}. Thus w(Alf A2) is a bounded subset of C 2. It is not known whether this set is convex, Dash [4, 6]. In this note, we shall show that there is faithful * representation of the C*-algebra generated by Ax and Ai9 C*(Aif A2), under which the joint numerical range of At and A2 is convex. Following Berberian and Orland [1], we study the joint numerical status of Ax and A2, Σ(Aλ, A2) = {{φ(Ax)9 Φ(A2)) such that φ is a state on C*(A19 A2)}. If Ax and A2 are commuting normal operators then Σ(Aif A2) — w(A19 A2). We also show that certain points in the boundary of w(A19 A2) are joint approximate reducing eigenvalues. For the sake of notational convenience, all the results are being stated for two commuting operators. However, the results hold for any finite family of commuting operators. Let B(H) denote the algebra of all bounded linear operators on

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