Structure Theorem for a -compact Operators

A contraction Tdefined on a complex Hilbert space is called Acompact if there exists a nonzero function/analytic in the open unit disc and continuous on the closed disc such that f( T) is a compact operator. In this paper, the factorization of / is used to obtain a structure theorem which describes the spectrum of T. Introduction. A bounded linear operator T on a complex Banach space X is called polynomially compact if there is a nonzero polynomial p(z) such that p(T) is compact. The spectrum of polynomially compact operators T has been completely described by F. Gilfeather [4] by showing that T is a finite sum of power compact operators. A similar problem of describing the spectrum arises in case p(z) is replaced by the uniform limit of polynomials. More explicitly, let T be a contraction defined on a Hilbert space, and A be the uniform closure of polynomials in the supremum norm over D; D is the open unit disc. The elements of A are analytic functions in D which have continuous extension to D. Forf(z) = 2 a„z" G A,f(T) = 2 a„T" converges in norm, and ||/(T)|| < H/lb. If / G H°°,fr(z) = f(rz) belongs to A for each r, 0 < r < 1, and \imr^x_0fr(T) exists in the norm. Define/(T) = limr_,x_0fr(T). For/ G A, this definition is consistent. The contraction T is called A-compact if there exists a nonzero/ G A such that/(T) is a compact operator. The problem is to describe the spectrum of ,4-compact contractions. Let T be an ,4-compact contraction on a complex Hilbert space. Denote by J(T) the set of functions/ G A such that/(T) is compact. It is easily seen that J(T) is a nonzero closed ideal of A, indeed, it is a principal ideal [5, Corollary, p. 88]. Let Z be the set of common zeros of functions in J(T) in D. Every function which generates J(T) is of the form/ = Eg where F is the greatest common divisor of the inner parts of functions in J(T) and g is the outer part of any function which belongs to A and vanishes precisely on Z D {\z\ = 1}. A generator/of J(T) is called a minimal function of T if its singularities which lie on the unit circle are contained in Z. A minimal function / always exists. An operator T defined on a Hilbert space X is said to be decomposed if there exists a pair (XX,X2) of subspaces of X such that X has unique decomposition in (XX,X2) and Xt is invariant under T, i = 1, 2. A decomposition of Tcan be obtained by the decomposition theorem [7, p. 419] if a(T) has more than one Received by the editors August 29, 1972 and, in revised form, July 15, 1975. AMS (MOS) subject classifications (1970). Primary 47A65, 47B99.