Browsing by Author "Klimek, Slawomir"
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Item Commutants of composition operators on the Hardy space of the disk(2013-11-06) Carter, James Michael; Cowen, Carl C.; Klimek, Slawomir; Perez, Rodrigo A.; Chin, Raymond; Bell, Steven R.; Mukhin, EvgenyThe main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.Item D-bar and Dirac Type Operators on Classical and Quantum Domains(2012-08-29) McBride, Matthew Scott; Klimek, Slawomir; Cowen, Carl C.; Ji, Ron; Dadarlat, MariusI study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.Item Derivations and Spectral Triples on Quantum Domains I: Quantum Disk(2017) Klimek, Slawomir; McBride, Matt; Rathnayake, Sumedha; Sakai, Kaoru; Wang, Honglin; Mathematical Sciences, School of ScienceWe study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.Item Dirac type operators on the quantum solid torus with global boundary conditions(Elsevier, 2020-04) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of ScienceWe define a noncommutative space we call the quantum solid torus. It is an example of a noncommutative manifold with a noncommutative boundary. We study quantum Dirac type operators subject to Atiyah-Patodi-Singer like boundary conditions on the quantum solid torus. We show that such operators have compact inverse, which means that the corresponding boundary value problem is elliptic.Item Noncommutative geometry of the quantum disk(Springer, 2022) Klimek, Slawomir; McBride, Matt; Peoples, J. Wilson; Mathematical Sciences, School of ScienceWe discuss various aspects of the noncommutative geometry of a smooth subalgebra of the Toeplitz algebra. In particular, we study the structure of derivations on this subalgebra.Item A Note on Dirac Operators on the Quantum Punctured Disk(National Academy of Science of Ukraine, 2010-07-16) Klimek, Slawomir; McBride, Matt; Mathematical Sciences, School of ScienceWe study quantum analogs of the Dirac type operator −2z¯¯¯∂∂z¯¯¯ on the punctured disk, subject to the Atiyah–Patodi–Singer boundary conditions. We construct a parametrix of the quantum operator and show that it is bounded outside of the zero mode.Item A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere(2014-03) Klimek, Slawomir; McBride, MattThe goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.Item A note on spectral properties of the p-adic tree(AIP, 2016) Klimek, Slawomir; Rathnayake, Sumedha; Sakai, Kaoru; Department of Mathematical Sciences, School of ScienceWe study the spectrum of the operator D∗D, where the operator D, introduced in Klimek et al. [e-print arXiv:1403.7263v2], is a forward derivative on the p-adic tree, a weighted rooted tree associated to ℤp via Michon’s correspondence. We show that the spectrum is closely related to the roots of a certain q − hypergeometric function and discuss the analytic continuation of the zeta function associated with D∗D.Item A Note on Spectral Triples on the Quantum Disk(National Academy of Science of Ukraine, 2019) Klimek, Slawomir; McBride, Matt; Peoples, John Wilson; Mathematical Sciences, School of ScienceBy modifying the ideas from our previous paper [SIGMA 13 (2017), 075, 26 pages, arXiv:1705.04005], we construct spectral triples from implementations of covariant derivations on the quantum disk.Item Restrictions to Invariant Subspaces of Composition Operators on the Hardy Space of the Disk(2014-01-29) Thompson, Derek Allen; Cowen, Carl C.; Ji, Ronghui ; Klimek, Slawomir; Bell, Steven R.; Mukhin, EvgenyInvariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.