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.