ANALYTIC FAMILIES OF COMPACT OPERATORS COMMUTING WITH THEIR DERIVATIVE

Spectral properties of analytic families of compact operators on a Hilbert space are studied. The results obtained are then used to establish that an analytic family of self-adjoint compact operators on a Hilbert space H, which commute with their derivative, must be functionally commutative. In [2], Stuart Goff studied analytic hermitian function matrices which commute with their derivative on some real interval I , i.e, A(t)A (t) = A (t)A(t) for all t ∈ I . He obtained as a main result that these matrices are
0
0
functionally commutative on I , i.e.,
A(s)A(t) = A(t)A(s)
for all s, t ∈ I [2], Theorem 3.6. Our aim is to further extend the result of Goff from matrices to the infinite- dimensional situation of compact self-adjoint operators on a separable Hilbert space. We study first analytic families of compact self-adjoint operators on a complex Hilbert space, which commute with their derivative on some real interval I . Our main result establishes that these operators must be functionally commutative on I , that is,
A(s)A(t) = A(t)A(s)
for all s, t ∈ I , extending the main result of [2] from the case of matrices to the infinite dimensional situation of operators on a Hilbert space. Then, we will explain how to solve the general problem when we consider only self-adjoint operators on a separable Hilbert space (without the compactness hypothesis), and comment on the other more general extension of our result to Analytic families of compact operators on a Banach space.