When is the spectrum of a convolution operator on \(L^ p\) independent of p? (English) Zbl 0715.47004
Let G be a locally compact topological group with a fixed left Haar measure and let \(f\in L^ 1(G)\). The main result proved is that the convolution operators defined on \(L^ p(G)\) by \((T_ f)_ p(g)=f*g\) have the same spectrum for all \(p\in [1,\infty]\) if and only if G is amenable and the Banach *-algebra \(L^ 1(G)\) is symmetric.
Reviewer: W.D.Evans
MSC:
47A10 | Spectrum, resolvent |
43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
47B38 | Linear operators on function spaces (general) |
Keywords:
locally compact topological group with a fixed left Haar measure; convolution operators; amenable; Banach *-algebraReferences:
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