Matrix and B*-algebra Eigenfunction Expansions

Abstract

<p>Let Mn be the set of all n by n complex matrices and [a,b] be a finite real interval. Let L²(a,b;Mn) be the set of all functions f: [a,b]→Mn such that all entry functions belong to L²(a,b). Under the usual addition, scalar multiplication, and inner-product defined by (F,G) = ∫ᵇa π(G*(t)F(t))dt, L²(a,b;Mn) is a Hilbert space. Consider the differential operator L defined by [equation removed], where the superscript (s) denotes s-times differentiation with respect to t, Pi and Y are n by n matrix functions and the domain of L is a linear subspace S of L²(a,b,Mn). Under very general hypotheses on the coefficients Pi, and an appropriate domain S given by end-point conditions, the operator L is symmetric, with finite and equal deficiency indices. The self-adjoint extensions L' of L are explicitly obtained and the following result holds: The self-adjoint operator L' has only a point spectrum, all its eigenvalues are of finite multiplicity, and every finite interval contains only a finite number of them.</p> <p>The classical Inversion Theorem is also generalized to obtain a unitary mapping from L²(a,b,Mn) onto L²σ, where σ is an appropiate matrix distribution function.</p> <p>Let now A be an H-algebra, that is a B-algebra for which there exists a positive linear functional π satisfying π(x*x) ≥ 0 and π(x*x) = 0 if and only if x = 0 for all x ε A. Then the following result holds: Let A be an L²-separable H-algebra, [a,b] a finite interval and q(x) an A-valued, continuous function on [a,b] with spectrum σ = σ(q(x)) such that inf {σ(q(x)): x ε [a,b]} is finite. Then the problem y" + (λ - q(x))y = 0, y(a,λ)cosμ - y'(a,λ)sinμ = 0, 0 ≤ μ ≤ π/2, y(b,λ) = 0, has a non-trivial solution for only a countable number of real values λi of λ, and if zi is the solution corresponding to λi, then for any A-valued, twice continuously differentiable function g(x) on [a,b] satisfying the above boundary conditions, we have g(x) = Σi (g,zi)zi(x). An approximation theorem in the singular case for functions on (-∞,∞) is also given.</p>