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unpublished notes of Krein's lecture in which, in particular, the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."
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