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Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

Iakovlev S. I., Iakovleva V.

Opuscula Mathematica

2013

Vol. 33, no. 1

81-98

It is shown that any μ ∈ C is an infinite multiplicity eigenvalue of the Steklov smoothing operator Sh acting on the space [formula]. For μ ≠ 0 the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.