This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities i...

We continue the analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and, following Sahlmann’s ideas [1], define a holonomy-flux ∗-algebra whose elements correspond to the elementary variables. There exists a natural action of autom...

Smooth functions are associated with operators on Hilbert spaces of analytic through the Berezin transform. The symbol and number an operator A functional space H(Ω) over some set Ω reproducing kernel defined, respectively, by 
 ̃(μ)=〈A K_μ/K_μ ,K_μ/K_μ 〉,μ∈Ω ber(A)=sup┬(μ∈Ω)⁡|A ̃(μ)|. By using this bounded function ̃, we present new inequalities operators. Some respect to Specht's ratio impr...

In this paper, we introduce a generalization of Hilbert $C^*$-modules which are pre-Finsler modules, namely, $C^{*}$-semi-inner product spaces. Some properties and results of such spaces are investigated, specially the orthogonality in these spaces will be considered. We then study bounded linear operators on $C^{*}$-semi-inner product spaces.

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functio...

Coordinate formalism on Hilbert manifolds developed in [1] is reviewed. The results of [1] are applied to the simpliest case of a Hilbert manifold: the abstract Hilbert space. In particular, functional transformations preserving properties of various linear operators on Hilbert spaces are found. Any generalized solution of an arbitrary linear differential equation with constant coefficients is ...