Abstract
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalized eigenvectors and conditions implying complete indeterminacy are also provided.
Highlights
A nonzero sequence will be called a generalized eigenvector associated with z ∈ C if it satisfies the recurrence relation an∗−1un−1 + bnun + anun+1 = zun, (n ≥ 1)
For the simplification of the condition for C(λ), we assume that the sequence tends to infinity; i.e., lim n→∞
This section is devoted to showing the implications of the nondegeneracy of (Qz : z ∈ ) together with the positivity of |Sn| to the asymptotics of the generalized eigenvectors
Summary
The operator A is called a block Jacobi matrix It is self-adjoint provided the Carleman condition is satisfied, i.e. n=0 an where · is the operator norm (see [2, Theorem VII-2.9]). The first main result of this article is Theorem 4, which generalizes the results obtained in [26] to the operator case. The method of the proofs of the presented theorems is based on an extension of the techniques used in [26,28] In these articles, one examines the positivity or the convergence of sequences of quadratic forms on R2 acting on the vector of two consecutive values of a generalized eigenvector u associated with λ ∈ ⊂ R; i.e., The real part of the operator.
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