Abstract

Introduction. It has been known for some time that on a complex infinite-dimensional Hilbert space there exist invertible operators without square roots, indeed without roots of any order, which therefore do not belong to the range of the exponential function. A first class of examples of such operators was described by Halmos, Lumer, and Schaffer [3]: the space considered was the separable Hilbert space of all complex-valued functions defined, analytic, and squaresummable on a domain D of the complex plane (with the L2-norm); the operator was the analytic position operator A defined by (A+5) (z) =zcf(z), zCD. It was shown that A is invertible and lacks a square root (indeed, a root of any order) if and only if D surrounds the origin but does not contain it. Halmos and Lumer [2] used the concept of multiplicity to show that the analytic position operator for such D is an interior point (in the norm topology for operators) of the set of invertible operators without roots. Recently, Deckard and Pearcy [1] described another class of invertible square-root-less operators on a separable Hilbert space; the spectral properties of these operators are quite different from those of the operators discussed in [3] and [2]. We shall not be concerned with this class of examples in this paper. Root-less operators are beginning to be used in various contexts: we mention, in particular, the disproof by Massera and Schaffer [5, p. 92], of an extension to Hilbert space of Floquet's Theorem on periodic differential equations. It appears that more information about such operators and more flexibility in their choice is desirable. To take the analytic position operator, for instance: there is of course some latitude in the choice of the domain D, but the description of the operator in terms of an orthonormal basis is awkward even in the simplest case, when D is an annulus centred at the origin, and quite unmanageable in other cases; besides, this description can not be perturbed or altered in any prescribed way as the application in hand might require. We remark, however, that the original definition has the advantage of making almost obvious the spectral structure that ensures the lack of roots.

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