COMPACTNESS AND BEREZIN SYMBOLS

Description

COMPACTNESS AND BEREZIN SYMBOLS I. CHALENDAR, E. FRICAIN, M. GURDAL, AND M. KARAEV Abstract. We answer a question raised by Nordgren and Rosen- thal about the Schatten-von Neumann class membership of oper- ators in standard reproducing kernel Hilbert H spaces in terms of their Berezin symbols. 1. Introduction Let ? be a subset of a topological space X such that the boundary ∂? is non-empty. Let H be an infinite dimensional Hilbert space of functions defined on ?. We say that H is a reproducing kernel Hilbert space if the following two conditions are satisfied: (i) for any ? ? ?, the functionals f 7?? f(?) are continuous on H; (ii) for any ? ? ?, there exists f? ? H such that f?(?) 6= 0. According to the Riesz representation theorem, the assumption (i) im- plies that, for any ? ? ?, there exists k? ? H such that f(?) = ?f, k??H, f ? H. The function k? is called the reproducing kernel of H at point ?. Note that by (ii), we surely have k? 6= 0 and we denote by k? the normalized reproducing kernel, that is k? = k?/?k??H. Following the definition of [NR94], we say that a reproducing kernel Hilbert space H is standard if k? ? 0 (weakly) as ? ? ?, for any point ? ? ∂?.

• let ?


• von neumann


• orthonormal sequences


• standard reproducing kernel


• key words


• lim n?


• now let


• berezin symbols


• point ?

Sujets

• Anderson
• Karayev
• John von Neumann

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 20
• Langue: English

Extrait

COMPACTNESSANDBEREZINSYMBOLS I.CHALENDAR,E.FRICAIN,M.GU ¨ RDAL,ANDM.KARAEV Abstract. WeansweraquestionraisedbyNordgrenandRosen- thalabouttheSchatten-vonNeumannclassmembershipofoper- atorsinstandardreproducingkernelHilbert H spacesintermsof theirBerezinsymbols. 1. Introduction LetΩbeasubsetofatopologicalspaceXsuchthattheboundary ∂Ωisnon-empty.Let H beaninfinitedimensionalHilbertspaceof functionsdefinedonΩ.Wesaythat H isareproducingkernelHilbert spaceifthefollowingtwoconditionsaresatisfied: (i)foranyλ ∈ Ω,thefunctionalsf 7−→ f(λ)arecontinuouson H ; (ii)foranyλ ∈ Ω,thereexistsf λ ∈H suchthatf λ (λ) 6 =0. AccordingtotheRieszrepresentationtheorem,theassumption(i)im- pliesthat,foranyλ ∈ Ω,thereexistsk λ ∈H suchthat f(λ)= h f,k λ i H ,f ∈H . Thefunctionk λ iscalledthereproducingkernelof H atpointλ.Note thatby(ii),wesurelyhavek λ 6 =0andwedenotebyk ˆ λ thenormalized reproducingkernel,thatisk ˆ λ =k λ / k k λ k H . Followingthedefinitionof[NR94],wesaythatareproducingkernel Hilbertspace H isstandardifk ˆ λ *0(weakly)asλ → ζ,forany pointζ ∈ ∂Ω.In[NR94],E.NordgrenandP.Rosenthalestablisheda characterizationofcompactoperatorsactingonsuchspacesinterms oftheBerezinsymbolsoftheirunitaryorbits.Recallthatif L ( H ) denotesthespaceoflinearandboundedoperatorson H ,thenthe BerezinsymbolT ˜ ofanyoperatorT ∈L ( H )isthefunctiondefinedon 2010MathematicsSubjectClassification.Primary47B38,4B07;Secondary 47B35. Keywordsandphrases.Berezinsymbols,compactoperators,Schatten-vonNeu- mannclasses,reproducingkernelHilbertspace,modelspaces. ThisworkwassupportedbythePHCBosphore2010-2012. 1