On the Four Block Problem, I

This paper is concerned with the study of the singular values of a “four block operator” which naturally appears in control engineering and which possesses a number of interesting mathematical properties. The study of this operator will be shown to be reducible to a skew Toeplitz operator problem of the kind studied in [1]. The main theoretical fact proven here is an explicit closed form formula for the essential norm of the four block operator (see Proposition 1.1). We dedicate this paper to the memory of Constantin Apostol who was a great master of the essential properties of operators in Hilbert space.