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:
| [1] | Ransford, Pacific J. Math. 121 pp 445– (1986) · Zbl 0546.46062 · doi:10.2140/pjm.1986.121.445 |
| [2] | Pytlik, Bull. Acad. Polon. Sci. Ser. Sci. Math. 21 pp 899– (1973) |
| [3] | Pier, Amenable Locally Compact Groups (1984) · Zbl 0597.43001 |
| [4] | Rickart, Banach Algebras (1960) |
| [5] | DOI: 10.1007/BF01418936 · Zbl 0264.43007 · doi:10.1007/BF01418936 |
| [6] | Bonsall, Complete Normed Algebras (1973) · doi:10.1007/978-3-642-65669-9 |
| [7] | DOI: 10.1007/BF01262044 · Zbl 0468.47022 · doi:10.1007/BF01262044 |
| [8] | Jörgens, Linear Integral Operators (1982) |
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