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Abstract
We show that the total multiplicities of negative and positive spectra of a self-adjoint Hankel
operator
H in
L 2 ( ℝ + ) with integral
kernel
h ( t )
and of the operator of multiplication by the inverse Laplace transform of
h ( t ) , the distribution
σ ( λ ) , coincide. In
particular,
±
H
≥ 0
if and only if
±
σ ( λ )
≥ 0 .
To construct
σ ( λ ) ,
we suggest a new method of inversion of the Laplace transform in appropriate
classes of distributions. Our approach directly applies to various classes
of Hankel operators. For example, for Hankel operators of finite rank, we
find an explicit formula for the total numbers of their negative and positive
eigenvalues.
Keywords
Hankel operators, convolutions, necessary and sufficient
conditions for positivity, sign function, operators of
finite rank, the Carleman operator and its perturbations
Mathematical Subject Classification 2010
Primary: 47A40
Secondary: 47B25
Milestones
Received: 13 April 2014
Accepted: 26 November 2014
Published: 15 April 2015