Extended resolvents and extended spectral functions of a Hermitian operator

In this paper we construct the theory of extensions of Hermitian operators which are initially defined on a manifold in Hilbert space. The operators may have infinite defect numbers, and the manifold may fail to be dense. The extension is accompanied by a result in the Hilbert space $\mathfrak H_-$ of ideal elements (generalized functions which are defined on the Hilbert space of elements which belong to the basic Hilbert space: $\mathfrak H_+\subset\mathfrak H$). We conduct a detailed analysis of extended generalized resolvents and corresponding spectral functions. We explain the connection between functions of the form $(\widehat R_\lambda f, f)$, where $\widehat R_\lambda$ is an extended generalized resolvent, and the theory of $R$-functions.
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