On continuity of linear transformations commuting with generalized scalar operators in Banach space

Pavla Vrbová

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The relationship between the joint spectrum γ(A) of an n-tuple $A = (A_{1}, . . ., A_{n})$ of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators $A_{j}$ mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact...