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Journal of Operator Theory

Volume 65, Issue 1, Winter 2011  pp. 211-232.

Multiplicities, generalized Jacobi matrices, and symmetric operators

Authors:  Bela Nagy
Author institution: Department of Analysis, Institut of Mathematics, Budapest University of Technology and Economics, Budapest, H-1521 Hungary

Summary:  The global multiplicity of {\it bounded} linear operators in Banach spa\-ces has been studied for a number of classes of operators. We introduce a definition of multiplicity of a {\it general unbounded} operator, and compare it with a known version (essentially reducing it to bounded cases) for certain symmetric operators. We study the connection of this concept with generalized (regular or irregular, block) Jacobi matrices. We establish the multiplicities of pure maximal symmetric operators, and show how this reveals the structure of the elementary symmetric operators and their simplest matrix representations: a problem unsolved in a classical paper by von Neumann.

Keywords:  global multiplicity, multicyclicity, (pure) maximal symmetric operator, elementary symmetric operator, deficiency index, infinite generalized (regular or irregular, Hermitian symmetric) Jacobi matrix, matrix representation of a symmetric operator

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