Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Studies on the Spectral Schwartz Distribution   The resolvent function of an operator in a Banach space is defined on an open
subset of the complex plane and is holomorphic. It obeys the resolvent
equation. A generalization of this equation to Schwartz distributions is
defined and a Schwartz distribution, which satisfies that equation is called a
resolvent distribution. In important cases the resolvent distribution is the
whole plane. Its restriction to the subset, where it is continuous, is the
usual resolvent function. Its complex conjugate derivative is ,but a factor,
the spectral Schwartz distribution, which is carried by a subset of the
spectral set of the operator. The spectral distribution yields a spectral
decomposition. We have a generalized orthogonality relation. Completeness is
defined in a natural way and is the case e.g. if the operator is bounded. The
spectral distribution of a matrix and a unitary operator are given. If the the
operator is a self-adjoint operator on a Hilbert space, the spectral
distribution is the derivative of the spectral family. We calculate the
spectral distribution and the eigen value problem of the multiplication
operator and of the multiplication operator with a non symmetric rank one
perturbation. The operator is not normal and may have discrete real or
imaginary eigenvalues or a nontrivial Jordan decomposition.

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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: cs
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con