M.G. Krein’s Lectures on Entire Operators

This book is devoted to the theory of entire Hermitian operators, an important branch of functional analysis harmoniously combining the methods of operator theory and the theory of analytic functions. This theory anables various problems of classical and modern analysis to be looked at from a uniform point of view. In addition, it serves as a source for setting and solving many new problems in both theories. The three chapters of the book are based on the notes written by his students of M. G. Krein's lectures on the theory of entire operators with (1,1)­ deficiency index which he delivered in 1961 at the Pedagogical Institute of Odessa, and on his works on the extension theory of Hermitian operators and the theory of analytic functions. The theory is further developed in the direction of solving the problems set up by Krein at ICM-66 in the first two appendices. The first concerns the case of Hermitian operators with arbitrary defect numbers, entire with respect to an ordinary gauge and to a generalized one as well. The other focuses on the entire operators representable by differential operators. The third appendix is the translation from Russian of the unpublished notes of Krein's lecture in which, in particular, the place of the theory of entire operators in the whole analysis is elucidated. In Krein's mathematical heritage the theory of entire operators occupies a special position.

1 Some aspects of operator theory and the theory of analytic functions.- 1 On spectral functions of Hermitian operators.- 2. Representation of a simple Hermitian operator whose deficiency index is (1,1).- 3. On the spectrum of self-adjoint extensions of a Hermitian operator whose deficiency index is (1,1).- 4. Some aspects of the theory of analytic functions.- 2 Entire operators whose deficiency index is (1,1).- 1. Quasiregular gauge.- 2. Principal theorem on regular points of fM(z).- 3. u -regular points of a Hermitian operator.- 4. Functions e(z) and ?M(Z).- 5. Entire operators.- 6. R-functions and generalized resolvents of a Hermitian operator.- 7. On distribution functions of a Hermitian operator.- 8. Method of directing functionals.- 9. Operator, entire with respect to a generalized gauge.- 3 Applications of entire operator theory to some classical problems of analysis.- 1. Power moment problem.- 2. Continuation problem for positive definite functions.- 3. Unitary and spiral curves.- Appendix 1 On entire operators whose defect numbers are arbitrary.- Appendix 2 Enture operators represented by differential operators.- Appendix 3 On a difficult problem of operator theory and its relation to classical analysis.- Comments.