Spectral Analysis of Operator Polynomials and Second-Order Differential Operators
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tral Analysis of Operator Polynomials and Second-Order Differential Operators A. G. Baskakov1* and D. B. Didenko1** 1
Voronezh State University, Voronezh, 394018 Russia
Received November 15, 2019; in final form, November 15, 2019; accepted April 23, 2020
Abstract—Studying spectral properties of operator polynomials is reduced to studying the corresponding spectral properties of operators defined by operator matrices. The results are used to investigate second-order differential operators by associating them with the corresponding first-order differential operators and using their properties related to invertibility. DOI: 10.1134/S0001434620090205 Keywords: bounded linear operator, differential operator, invertibility states, spectrum, Fredholm operator, projection operator.
1. MAIN RESULTS Let X and Y be complex Banach spaces, and let Hom(X , Y ) be the Banach space of bounded linear operators (homomorphisms) defined on X and ranging in Y . End X = Hom(X , X ) is the Banach algebra of endomorphisms of the space X . Let A : D(A) ⊂ X → X be a linear operator with nonempty resolvent set ρ(A). A linear operator A : D(A ) = D(AN ) ⊂ X → X of the form A = B0 AN + B1 AN −1 + · · · + BN , where B0 , B1 , B2 , . . . , BN −1 , BN , N ∈ N, belong to the algebra End X , is called an operator polynomial (of order N with operator coefficients B0 , B1 , . . . , BN expanded in a power series in the operator A). For a bounded operator A ∈ End X , operator polynomials were considered in [1]–[3], where they were applied to study difference operators. Invertibility problems for second-order operator polynomials with an unbounded operator A were considered in [4] (mainly for second-order differential operators acting on a space of functions defined on the whole real axis). Estimates for bounded solutions of higher-order differential equations were obtained in [5]. In this paper, we consider second-degree operator polynomials A = B0 A2 + B1 A + B2 : D(A2 ) ⊂ X → X
(1.1)
with domain D(A ) = D(A2 ) = {x ∈ D(A) : Ax ∈ D(A)}, where Bk ∈ End X , k = 0, 1, 2, and the operator B0 is not required to be invertible. Along with the operator A , we define an operator A : D(A) ⊂ X × X → X × X , by using the operator matrix
A∼
* **
A B2
D(A) = D(A) × D(A),
−I ; B0 A + B1
E-mail: [email protected] E-mail: [email protected]
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BASKAKOV, DIDENKO
namely, A(x1 , x2 ) = (Ax1 − x2 , B2 x1 + B0 Ax2 + B1 x2 ),
(x1 , x2 ) ∈ D(A) = D(A) × D(A).
The operator A can be written as A = B0 A0 + B1 , where the operators B0 and B1 belong to the Banach algebra End(X × X ), A0 : D(A) × D(A) ⊂ X × X → X × X , and these operators are defined by the matrices I 0 A 0 0 −I , B1 ∼ , A0 ∼ . B0 ∼ 0 B0 0 A B2 B1 Definition 1. Let B : D(B) ⊂ Y → Z be a closed linear operator. We consider the following conditions: (1) the kernel Ker B = {y ∈ D(B) : By = 0} of the operator B is the zero subspace of Y (the operator B is injective); (2) the dimension dim Ker B = n of the kernel Ker B of the operator B is positive and finite; (3) Ker B is an infinite-dimensional
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