In this note, we show that the set of all commuting d-tuples of commuting n × n matrices that are contained is an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also stu...

We consider the problem of counting simultaneous conjugacy classes n-tuples in a finite group G. Let αG,n denote number n-tuples, and βG,n commuting study generating functions AG(t)=∑n≥0αG,ntn, BG(t)=∑n≥0βG,ntn, which are rational t. show that AG(t) determines is completely determined by class equation prove normalized AG(t/|G|) BG(t/|G|) invariants isoclinism families. compute exponential grow...

Abstract. We generalize earlier results of [2], [3], [6], [13], [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B ∞,1(R ), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then ‖f(A1 · · · , An)− f(B1, · · · , Bn)‖ ≤ constmax1≤j≤n ‖Aj −Bj‖ α for all n...

In the paper one considers the local structure of the Fredholm joint spectrum of commuting n-tuples of operators. A connection between the spatial characteristics of operators and the algebraic invariant of the corresponding coherent sheaves is investigated. A notion of Weyl joint spectrum of commuting n-tuple is introduced.