The Experts below are selected from a list of 24 Experts worldwide ranked by ideXlab platform
A. K. Motovilov - One of the best experts on this subject based on the ideXlab platform.
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Solvability of the Operator Riccati Equation in the Feshbach Case
Mathematical Notes, 2019Co-Authors: S. Albeverio, A. K. MotovilovAbstract:Let L be a bounded 2 × 2 block operator matrix whose Main-Diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other Main-Diagonal Entry is entirely contained in this band. We establish conditions under which the operator matrix L admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed L possess bounded solutions. The same conditions also ensure a Markus–Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.
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On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
Mathematical Notes, 2016Co-Authors: S. Albeverio, A. K. MotovilovAbstract:We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one Main-Diagonal Entry is embedded into the absolutely continuous spectrum of the other Main-Diagonal Entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the spectral parameter plane. We present the conditions under which the continued Schur complement has operator roots, in the sense of Markus-Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example
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On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
Mathematical Notes, 2016Co-Authors: S. Albeverio, A. K. MotovilovAbstract:We consider a J -self-adjoint 2 × 2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one Main-Diagonal Entry of L is embedded into the absolutely continuous spectrum of the other Main-Diagonal Entry. We work with the analytic continuation of the Schur complement of aMain-Diagonal Entry in L − z to the unphysical sheets of the spectral parameter z plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J -orthogonal invariant subspaces of L . The presentation ends with an explicitly solvable example.
S. Albeverio - One of the best experts on this subject based on the ideXlab platform.
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Solvability of the Operator Riccati Equation in the Feshbach Case
Mathematical Notes, 2019Co-Authors: S. Albeverio, A. K. MotovilovAbstract:Let L be a bounded 2 × 2 block operator matrix whose Main-Diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other Main-Diagonal Entry is entirely contained in this band. We establish conditions under which the operator matrix L admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed L possess bounded solutions. The same conditions also ensure a Markus–Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.
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On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
Mathematical Notes, 2016Co-Authors: S. Albeverio, A. K. MotovilovAbstract:We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one Main-Diagonal Entry is embedded into the absolutely continuous spectrum of the other Main-Diagonal Entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the spectral parameter plane. We present the conditions under which the continued Schur complement has operator roots, in the sense of Markus-Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example
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On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
Mathematical Notes, 2016Co-Authors: S. Albeverio, A. K. MotovilovAbstract:We consider a J -self-adjoint 2 × 2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one Main-Diagonal Entry of L is embedded into the absolutely continuous spectrum of the other Main-Diagonal Entry. We work with the analytic continuation of the Schur complement of aMain-Diagonal Entry in L − z to the unphysical sheets of the spectral parameter z plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J -orthogonal invariant subspaces of L . The presentation ends with an explicitly solvable example.
Chaoqian Li - One of the best experts on this subject based on the ideXlab platform.
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An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry
Symmetry, 2018Co-Authors: Weiqian Zhang, Chaoqian LiAbstract:A set to locate all eigenvalues for matrices with a constant Main Diagonal Entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant Main Diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.
Weiqian Zhang - One of the best experts on this subject based on the ideXlab platform.
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An Eigenvalue Inclusion Set for Matrices with a Constant Main Diagonal Entry
Symmetry, 2018Co-Authors: Weiqian Zhang, Chaoqian LiAbstract:A set to locate all eigenvalues for matrices with a constant Main Diagonal Entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant Main Diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.
Pauline Van Den Driessche - One of the best experts on this subject based on the ideXlab platform.
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Elementary biDiagonal factorizations
Linear Algebra and its Applications, 1999Co-Authors: Charles R. Johnson, Dale D. Olesky, Pauline Van Den DriesscheAbstract:Abstract An elementary biDiagonal (EB) matrix has every Main Diagonal Entry equal to 1, and exactly one off-Diagonal nonzero Entry that is either on the sub- or super-Diagonal. If matrix A can be written as a product of EB matrices and at most one Diagonal matrix, then this product is an EB factorization of A. Every matrix is shown to have an EB factorization, and this is related to LU factorization and Neville elimination. The minimum number of EB factors needed for various classes of n-by-n matrices is considered. Some exact values for low dimensions and some bounds for general n are proved; improved bounds are conjectured. Generic factorizations that correspond to different orderings of the EB factors are briefly considered.
