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The notions of norm and spectral radius of a matrix are generalized to spaces with an indefinite metric. A spectral radius formula is established.

This paper introduces a new class (denoted byEK) of operators acting on a direct sum ofm copies of l2. Inversion and Fredholm properties of operators inEK are studied. The classEK includes block Toeplitz operators with a rational matrix symbol and non-Toeplitz operators like operators defined by infinite block band matrices. Operators inEK are characterized by the following two properties: (1) the entriesajk in their canonical block matrix representation decay exponentially as functions of |j−k|, and (2) the Kronecker rank is finite. The main tool in the analysis is based on the fact that operators in the classEK can be represented as input-output operators of certain finite dimensional linear time-variant input-output systems. In the description and the study of these systems dichotomy of difference equations plays an important role.

Well known theorems concerning positive definite extensions of banded Toeplitz matrices are generalized. The proofs are based upon the theory ofm-sequences which is developed herein. The nonstationary versions of the maximum distance and maximum entropy problems are discussed.

In this formula i t is agreed that whenever the transformation (PnTPn-XI)IHn does not have an inverse we write H((PnTPn-XI)IHn)'III = +~ . Denote by ~a(T) the approximation spectrum of T and by o(T) the spectrum of T . I t is easily seen that Oa(T) is compact. In fact i t follows from the definit ion that X belongs to the complement of Oa(T) i f f there exists an in f in i te sequence of integers n I < n 2 < . . such that Hnk) II I supll((P. TP,~ X I ) I < ~o k "k '~ I f T is self-adjoint i t is clear that ~(T) C~a(T) . In this case we wil l use the following property of ~a(T) , ~ is in ~a(T) i f and }, ~ E O(PnTPnlH n) for each only i f there is a sequence { n}n=i such that 1n n and lim ~'n = X . In this paper we consider the following problem. When does the equality Oa(T) = ~(T) holds, T being a self-adjoint operator with a f in i te number of nonzero diagonals. We say that T is a 2n + 1 diagonal operator i f t i j = 0 for