The index of triangular operator matrices
Copyright policy )The index of triangular operator matrices
For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.
Banach lattices, index, triangular operator matrix, Eigenvalues, singular values, and eigenvectors, order of a pole of the resolvent, nonnegative operator, direct sum of complex Banach spaces, Positive linear operators and order-bounded operators, Banach lattice, Rothblum index theorem, Laurent series, Positive matrices and their generalizations; cones of matrices, Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators, Spectrum, resolvent
Banach lattices, index, triangular operator matrix, Eigenvalues, singular values, and eigenvectors, order of a pole of the resolvent, nonnegative operator, direct sum of complex Banach spaces, Positive linear operators and order-bounded operators, Banach lattice, Rothblum index theorem, Laurent series, Positive matrices and their generalizations; cones of matrices, Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators, Spectrum, resolvent
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