2018/02/26 by Glück, Jochen
#46B08 #46E30 #47B38 #47D06 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary: 47A10 #Secondary: 47B65 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1802.09422
We consider a positive and power-bounded linear operator T on Lp over a finite measure space and prove that, if TLp ⊆ Lq for some q > p, then the essential spectral radius of T is strictly smaller than 1. As a special case, we obtain a recent result of Miclo who proved this assertion for self-adjoint ergodic Markov operators in the case p=2 and thereby solved a long-open problem of Simon and Høegh-Krohn. Our methods draw a connection between spectral theory and the geometry of Banach spaces: they rely on a result going back to Groh that encodes spectral gap properties via ultrapowers, and on the fact that an infinite dimensional Lp-space cannot by isomorphic to an Lq-space for q \not= p. We also prove a number of variations of our main result: (i) it follows from theorems of Lotz and Martínez that the condition TLp ⊆ Lq can be replaced with the weaker assumption that T maps the positive part of the Lp-unit ball into a uniformly p-integrable set; (ii) while it is known that the positivity assumption on T cannot in general be omitted, we show that we can replace it with the assumption that T is contractive both on Lp and on Lq; (iii) we prove a version of the theorem which allows us, under appropriate circumstances, to also consider non-finite measures spaces; (iv) our result also has a uniform version: there exists an upper bound c ∈ [0,1) for the essential spectral radius of T, where c depends on certain quantitative properties of T, Lp and Lq.