(Ultra)differentiable functional calculus and current extension of the resolvent mapping

A functional calculus for unbounded generalized scalar operators on Banach spaces

“…The initial work of Dynkin [1972] was followed by the articles [Helffer and Sjöstrand 1989;Andersson 2003;Andersson and Sjöstrand 2004;Andersson et al 2006]. The results obtained by this approach are mostly consequences of our results.…”

“…The results obtained by this approach are mostly consequences of our results. In particular, Andersson [2003] showed that the functional calculus can be extended via the Cauchy-Green formula to the algebra of C ∞ functions with an additional condition at infinity if and only if the corresponding group is polynomially bounded, although a wider class of operators admits functional calculus for test functions. Andersson, Samuelsson and Sandberg [2006] observed that for any proper f ∈ C ∞ ‫ޒ(‬ m , ‫ޒ‬ k ), the set {g • f | g ∈ Ᏸ(‫ޒ‬ k )} is contained in D(‫ޒ‬ m ); they then used this fact to construct the functional calculus (with test functions) for f (A), where f is a proper C ∞ function and A is a Helffer-Sjöstrand operator, that is, an operator with real spectrum and rationally bounded resolvent on compacts.…”

Funkcionalni racuni za n-torke komutirajucih neogranicenih operatora

“…Prvi rezultati mogu se na i kod Dynkina u [15], a naredni u [18], [1], [3] i [4]. U Ƭima se funkcionalni raqun razvija na skoro holomorfnim raxireƬima test funkcija.…”

“…Treba re i da je ve ina ovih rezultata posledica rezultata ove disertacije. Zaista, u [1] je pokazano da je funkcionalni raqun uveden preko Koxi-Grinove formule, mogu e proxiriti do algebre C ∞ funkcija sa dodatnom osobinom ponaxaƬa u beskonaqnosti ako i samo ako je odgovaraju a grupa polinomno ograniqena, iako xira klasa operatora dopuxta funkcionalni raqun na test funkcijama. ZanimƩiv pristup moжe se na i i u [4].…”

“…U prvom odeƩku je upore en (ultra) slab funkcionalni raqun sa nekim drugim aksiomatskim pristupima; pokazano je da je ultra slab funkcionalni raqun opxtiji. Preciznije, dobijena je odgovaraju a teorija za xiru algebru funkcija (kako je pokazano u [1], klasa operatora se poklapa sa klasom koja se posmatra pri pristupu uvo eƬa funkcionalnog raquna imitiraƬem Koxi-Grinove integralne formule). U drugom odeƩku navedeni su neki primeri funkcionalnih raquna, a u tre em neki komentari i problemi koji mogu biti predmet daƩeg istraжivaƬa.…”