M.G. Krein’s Lectures on Entire Operators pp 47-118 | Cite as

# Entire Operators whose Deficiency Index is (1,1)

## Abstract

Chapter 2 deals mainly with the theory of entire Hermitian operators on a Hubert space ℌ, whose deficiency index is (1,1). To characterize the results in detail, recall that the assertions presented in the previous chapter lead to the conclusion that for every simple Hermitian operator A with defect numbers equal to 1 a one-dimensional subspace *M* (a fixed vector *u* ∈ *M* :∥*u* ∥ = 1 is called a gauge) can be found so that the original space ℌ is decomposed into the direct sum *M* ∔ ℌ= *(A — zI)D* (A)J for each non-real *z* except for at most a countable set whose limit points may be located only on the real axis. This decomposition generates the map Ф : *f* ↦ *fu* (z) from ℌ into a certain space of functions *fu(z* ) meromorphic inside the upper and lower half-planes. At the same time the operator A is transformed into the multiplication by the independent variable. It is established in section 1 that under the condition on the operator *A* that the set of all *f* for which the functions *fu* (*z* ) are analytic on the whole real axis is dense in ℌ (in this case the gauge *u* is called quasiregular), there exists a bounded non-decreasing function σ(λ) such that Ф is an isometry from ℌ into *L* _{2} (ℝ1, dσ). Such a function σ(λ) is called a distribution function associated with the gauge *u* of the operator *A* . It is not unique if the gauge is fixed. The description of all functions σ(λ) possessing this property is given in terms of spectral functions of the operator *A* . Those for which the space ℌ is isometrically isomorphic to L_{2}(ℝ1, dσ), are distinguished. Though section 2 is auxiliary in character, it is of interest in its own right. It is shown there that if the function *fu* (*z* ) is analytic in some neighbourhood ∣*z* — *a* ∣ < 2r of a point *a* ∈ ℂ1, then the inequality sup ∣*fu* (*z* )∣ ≤*c* ∥*f* ∥ holds; moreover, the constant *z* :∣*z* —*a* ∣<*r* c does not depend on the choice of *f* . The notion of a *u* -regular point of the operator *A* is introduced in section 3. Such a point is understood to be one at which all the functions *fu* (*z* ) are analytic. The relation of *u* -regular points of the operator A to its points of regular type is clarified. In view of the above estimate, *fu* (*z*) is a linear continuous functional on ℌ at any *u* -regular point of *A* . Therefore it admits a representation of the form The function *e* (*z* ) is investigated in section 4 . This function characterizes the inclination of the subspace *M* to. It plays an important role in problems like the classical moment problem. Section 5 is devoted to study of *M* -entire operators, that is the operators for which every function *fu* (*z* ) is entire. It is proved that in the case of entire operator, the growth of these functions is at most exponential. Their indicator diagrams are certain intervals of the imaginary axis, and the smallest interval on this axis containing them coincides with the indicator diagram of the log-subharmonic function. One of the principal results of this section consists of the fact that the spectrum of any self-adjoint extension within ℌ of an entire operator is discrete, and its eigenvalues are zeroes of a corresponding entire function of exponential type which is explicitly expressed through *e(z)* .This made it possible to write the asymptotic distribution formula for the eigenvalues of such extensions.

## Keywords

Real Axis Spectral Function Imaginary Axis Hermitian Operator Generalize Gauge## Preview

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