## Functions of A-Bounded Type in the Half-PlaneThis book is related to the theory of functions of a-bounded type in the ha- plane of the complex plane. I constructed this theory by application of the Li- ville integro-differentiation. To some extent, it is similar to M.M.Djrbashian's factorization theory of the classes Na of functions of a-bounded type in the disc, as much as the well known results on different classes and spaces of regular functions in the half-plane are similar to those in the disc. Besides, the book contains improvements of several results such as the Phragmen-Lindelof Principle and Nevanlinna Factorization in the Half-Plane and offers a new, equivalent definition of the classical Hardy spaces in the half-plane. The last chapter of the book presents author's united work with G.M. Gubreev (Odessa). It gives an application of both a-theories in the disc and in the half-plane in the spectral theory of linear operators. This is a solution of a problem repeatedly stated by M.G.Krein and being of special interest for a long time. The book is proposed for a wide range of readers. Some of its parts are comprehensible for graduate students, while the book in the whole is intended for young researchers and qualified specialists in the field. |

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a-bounded type application arbitrary assertion assume Ba(w belongs Blaschke type products boundary bounded Chapter characteristics coincides complete condition Consequently consider constant continuous convergent defined Definition depending difference disc domain du(t eigenvalue equality equivalent estimate exists factorization finite fixed follows formula function functions of a-bounded Further given gives half-plane hand Hence holds implies inequality integral known Lemma Let F(w lim inf limit log F(w measure meromorphic natural Nevanlinna nondecreasing nonnegative observe obtain obvious operator points Proof Proof of Theorem properties prove relation remains Remark representation requirements satisfies sequence set of zeros side similar statements subharmonic sufficiently Theorem 1.1 true uniformly valid values verify W-a log zeros