Ronveaux , Zeros of Gegenbauer and Hermite polynomials and connection coefficients , Mathematics of Computation , vol. Canuto, Y. Hussaini, A. Quarteroni, and T.
Zang , Spectral Methods in Fluid Dynamics , Ching-chau , Nonlinear Eigenvalues and Analytic-Hypoellipticity. Memoirs of the Amer , Math. Society , vol.
- Introduction to the Spectral Theory of Polynomial Operator Pencils.
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The Quadratic Numerical Range of an Analytic Operator Function | SpringerLink
Radzievskii and others. In these works under some additional conditions was proved the existence of two kind of the bases of the eigen and associated vectors of operator. Later, in the work  Allakhverdiev proved the existence of two kind of multiple completeness system of eigen and associated vectors of the operator pencil, rationally depending on parameter. Theorems about multiple expansions of the root subspaces of the operator pencil were proved in the works of RM Dzhabarzadeh , V.
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Vizitey, A. Through are designated the sequence of the characteristic values of the operator , in order of increasing various modules taking into account their multiplicities. In the work  it is proved the summation of the series on eigen and associated vectors of the operator pencil 1 by the method of Abel. The work  is devoted also to the questions of multiple summation of series on eigen and associated vectors of operator pencil 1 under the conditions that the resolvent of operator pencil on closed expanding indefinitely contours is uniformly bounded.
Below it is presented a theorem asserting the existing of the multiple basis of eigen and associated vectors of the operator pencil. The sequence of subspaces is called a basis of subspaces if the any vector of the space is expanded in the series by unique manner.
Preliminary Definitions. Let , , in 2 be completely continuous linear operators, acting in a separable Hilbert space. This element is called an eigenvector corresponding to the eigenvalue. There is an associated vector to the eigenvector if the following series of equations. Keldysh built the derivative systems according to the rule:. Linearly-independent elements form a chain of a set 5 of eigen and associated e. The study of the spectral properties of the equation in a Hilbert space is reduced to the study of the spectral properties of the equations and in the direct sum of copies of the Hilbert space.
And equation. Operators act in direct sum of copies of the Hilbert space. Consider the operator.
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Operator is the completely continuous in the space direct sum of copies of the space so the operators are completely continuous in the space. So the operator is a selfadjoint, completely continuous, then the characteristic values of lie on the n rays emanated from origin, norms of characteristic values of operators and coincide.
Suppose that the following conditions are satisfied:. Then the eigen values of the operator can be grouped so that the subspaces spanned by their eigen and associated vectors corresponding to the eigen values form -multiple basis with brackets in the space. All eigen values are divided on two sequences; one of them has the limit in infinity, other has the limit point in zero.
We evaluate the resolvent-operator for the values. Let then. The equalities. So , then starting with some. Denote though the domain of the complex plain bounded by contour and and though the contour, bounded the domain. It is known that the meaning of this integral is equal to the sum of residues respectively of all poles of integrand, lying within the domain. Let be a pole. Samenvatting The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra.
The second part is devoted to applications.
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Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators.
The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered.
precarexel.tk The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs.
Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader's background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed.
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