COMMUTABLE MATRICES AND FUNCTIONAL COMMUTATIVITY OF COMPACT NORMAL OPERATORS

Abstract. A simple expression is established for an analytic family of commutable matrix- valued functions. Then a characterization of two by two functional commutative matrices is proven. 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 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 normal 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.
Finally, it is shown that a family of analytic normal compact operators on a Hilbert space H, which commute with their derivatives, must be functionally commutative.
Keyword: Commutable matrix valued-function, Compact operator, Functional commutativ- ity, Normal operator, self-adjont operator, Riesz pro jection, spectral decomposition, analytic operator-valued function.
AMS 2010: Primary 47B15, Secondary 47A55