Spectral Components of Selfadjoint Block Operator Matrices with Unbounded Entries

Abstract

This paper is devoted to the study of the spectral components of selfadjoint operator matrices which are generated by symmetric operator matrices of the form in the product Hilbert space 'HI x ? i 2 where the entries A , B and C are not necessarily bounded operators in the Hilbert spaces 'HI, 312 or between them, respectively. Under suitable assumptions a selfadjoint operator L is associated with Lo and the spectral properties of L are studied. The main result concerns the case in which the spectra of the selfadjoint operators A and C are weakly separated. If a is a real number such that max u ( C ) 5 a 5 min u(A), descriptions of the spectral subspaces of L corresponding to the intervals ] -00, a ] and ]a, m[ and of the restrictions of L to these subspaces are given. From this main result half range completeness and basis properties for certain parts of the spectrum of L are deduced. The paper closes with two applications to systems of differential operators from magnetohydrodynamics. 0.In (0.1) 2 representations of the compressed resolvents of L are given which correspond to the first and second components in (0.1). Formally these compressed resolvents are the inverses of the Schur complements, however, the unboundedness of the operators A , B , C makes the formulas more involved. With these compressed resolvents also the corresponding compressed spectral functions of L are introduced. Starting from Section 3, the assumption that the spectra of A and C are separated is imposed. If a is as above, the kernel of La is described. These results are needed in Section 4 where by means of the Stieltjes inversion formula and the representations of the compressed resolvents of Section 2 the basic estimates of the compressed spectral functions are proved (Theorems 4.2 and 4.3). In Section 5 these estimates yield the angular operator representation of the spectral subspaces C -:= 4m,(l~ and C+ := C],,,[. This representation allows a description of the restrictions Llc-, LJL, as a perturbation of C in 3-12 and a perturbation of A in 711, respectively. In Sections 6 and 7 the spectral functions of these restrictions and the above mentioned half range completeness and basis properties are considered. Two applications to systems of ordinary and partial differential operators are considered in Sections 8 and 9.

Existence and Uniqueness of Contractive Solutions of Some Riccati Equations

“…In this case no assumption about B, in particular no smallness assumption on B, is needed. This result was extended in [3,13] and further in [14], where also a different method of proof was used. Namely, it is well known that an operator K from H 1 into H 2 is a contraction if and only if its graph subspace G 1 (K) (see Section 2) is maximal nonnegative with respect to the indefinite inner product Therefore the existence of a contractive solution of the Riccati equation (1.1) is equivalent to the existence of a maximal nonnegative invariant subspace of the Hamiltonian L in the Krein space H=H 1 ÄH 2 with this indefinite inner product.…”

Operator interpretation of the resonances generated by 2×2 matrix Hamiltonians

“…, where as previously where Γ l stands for a K B -bounded contour satisfying the condition (14). The operator Ω (l) does not depend (for a fixed l) on the choice of such a Γ l .…”

“…In the present work we use the standard definition of holomorphy of an operator-valued function with respect to the operator norm topology (see, e. g., [14]). One can extend to operator-valued functions the usual definitions of the spectrum and its components.…”

“…An only essential point is a checking of independence of the radius r 0 (B) of the multi-index l. To be sure in such an independence we consider an arbitrary ✷ So, for a given holomorphy domain D l the solutions X and H 1 , H 1 = A 1 + X, do not depend on the K B -bounded contours Γ l ⊂ D l satisfying the condition (14). But when the index l changes, X and H 1 can also change.…”

“…The mentioned conditions were then somewhat weakened by R. Mennicken and A. A. Shkalikov [15] in the case of a bounded entry A 1 and the same type of entries B ij as in [14]. Instead of the explicit conditions on the spectra of A i , the paper [15] uses a condition on the spectrum of the transfer function M 1 (z) itself.…”