Criteria for the Existence of Common Points of Spectra of Several Operator Pencils

In this paper we present two criteria for the existence of common eigen values of several operator pencils having discrete spectrum. One of the given criteria is proved by using analogs of resultant for several operator pencils; proof of the other criterion requires the use of the results of the multiparameter spectral theory. In both cases the number of operator pencils is finite, operator pencils act, generally speaking, in different Hilbert spaces.


Introduction
The definition of abstract analogue of resultant of two operator pencils in one parameter is given in [2] and [10] with the help of definitions of tensor product of spaces and tensor product of operators. When the operator pencils have the identical degree concerning parameter concept of resultant has been given in work of Khayniq [9], for operator pencils, generally speaking, with the different orders of parameter , the abstract analog of a resultant is studied by Balinskii [2].
Let be  (2) In a matrix 1 2 Re ( ( ), ( )) s A A λ λ the number of rows with operators i A is equal to leading degree of parameter λ in pencil ( ) B λ , that is m , the number of rows in matrix 1 2 Re ( ( ), ( )) s A A λ λ with operators i B coincides with the leading degree of parameter λ in pencil ( ) A λ , that is n .
Operator (2) At the proof of the second criterion of existence of common eigenvalue of several operator pencils in Hilbert space we essentially use the results of multiparameter spectral theory [1,3,5,11] and also the notion of abstract analog of Cramer's determinants.
Criterion for the existence of common eigenvalues of several operator pencils in Hilbert space In the matrix Proof of Theorem 2. Necessity. We suppose, that pencils , Then element X in the kernels of operators

H H
Further we use the known property of the elements of tensor product space. It is known that the representation of the element in tensor product space is not unique. For each element of the tensor product space there is the number coinciding with the minimal number of decomposable tensors, necessary for the representation of this element. This number is named the rank of element. If the sum decomposable tensors in the representation of element are more than the rank of this element then one-nominal components of the given element are linear dependent. Having transferred to each line of equalities (7) one decomposable tensor from left side in the right side of the this eqalities, we get, that the series standing at the left side in each equation have a rank 1 as they are equal to a decomposable tensor, standing in the right side of this equality. Thus, between one-nominal components of all terms entering into expression (7) there is a linear dependence and an element standing at the left in all equalities in (7)  we prove there is a number λ being the common point of Let the linear multiparameter system in the form be: ,...
We give a new criterion for the existence of common points of spectra of all operator pencils (5). Let us assume that each operator pencil in (5) has a discrete spectrum. Conditions on the operators where the operators are true. So on eigenvectors of the system   We use the known formulae of multiparameter spectral theory [1,3,11]    Criteria for the Existence of Common Points of Spectra of Several Operator Pencils (15).From [5] it follows that the system of eigen and associated vectors of multiparameter system (15)