Commutativity of compact selfadjoint operators
Studia Mathematica, Tome 112 (1995) no. 2, p. 109.

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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 operators, coincidence of γ(A) and supp (T(A)) no longer implies commutativity of the set A i .
Classification : 47A13, 47A25, 47A60, 47B15
Mots-clés : joint spectrum, Weyl calculus, non-compact operators 
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     author = {G. Greiner and W. Ricker},
     title = {Commutativity of compact selfadjoint operators},
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G. Greiner; W. Ricker. Commutativity of compact selfadjoint operators. Studia Mathematica, Tome 112 (1995) no. 2, p. 109. https://geodesic-test.mathdoc.fr/item/STUMA_1995__112_2_216141/