Eigenvalues of Integral Operators. II.
We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form , and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.
A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E, where r(·) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.
We characterize a class of *-homomorphisms on Lip⁎(X,𝓑(𝓗 )), a non-commutative Banach *-algebra of Lipschitz functions on a compact metric space and with values in 𝓑(𝓗 ). We show that the zero map is the only multiplicative *-preserving linear functional on Lip⁎(X,𝓑(𝓗 )). We also establish the algebraic reflexivity property of a class of *-isomorphisms on Lip⁎(X,𝓑(𝓗 )).