Möbius transformations and monogenic functional calculus

Kisil, Vladimir¹

¹ Institute of Mathematics, Economics and Mechanics, Odessa State University, ul. Petra Velikogo, 2, Odessa-57, 270057, Ukraine

Electronic research announcements of the American Mathematical Society, Tome 02 (1996) no. 1, pp. 26-33.

Résumé

A new way of doing functional calculi is presented. A functional calculus $\Phi : f(x)\rightarrow f(T)$ is not an algebra homomorphism of a functional algebra into an operator algebra, but an intertwining operator between two representations of a group acting on the two algebras (as linear spaces). This scheme is shown on the newly developed monogenic functional calculus for an arbitrary set of non-commuting self-adjoint operators. The corresponding spectrum and spectral mapping theorem are included.

Détail

DOI : 10.1090/S1079-6762-96-00004-2

Bibliographie

[1] Ahlfors, Lars V. Möbius transformations in 𝑅ⁿ expressed through 2×2 matrices of Clifford numbers Complex Variables Theory Appl. 1986 215 224

[2] Anderson, Robert F. V. The Weyl functional calculus J. Functional Analysis 1969 240 267

[3] Brackx, F., Delanghe, Richard, Sommen, F. Clifford analysis 1982

[4] Curto, R. E., Vasilescu, F.-H. Automorphism invariance of the operator-valued Poisson transform Acta Sci. Math. (Szeged) 1993 65 78

[5] Delanghe, R., Sommen, F., Souček, V. Clifford algebra and spinor-valued functions 1992

[6] Helemskii, A. Ya. Banach and locally convex algebras 1993

[7] Kirillov, A. A. Elements of the theory of representations 1976

[8] Mcintosh, Alan, Pryde, Alan A functional calculus for several commuting operators Indiana Univ. Math. J. 1987 421 439

[9] Reed, Michael, Simon, Barry Methods of modern mathematical physics. I 1980

[10] Maclane, Saunders Steinitz field towers for modular fields Trans. Amer. Math. Soc. 1939 23 45

[11] Rota, Gian-Carlo, Strang, Gilbert A note on the joint spectral radius Indag. Math. 1960 379 381

[12] Takesaki, Masamichi Structure of factors and automorphism groups 1983

[13] Taylor, Joseph L. The analytic-functional calculus for several commuting operators Acta Math. 1970 1 38

[14] Taylor, Joseph L. A general framework for a multi-operator functional calculus Advances in Math. 1972 183 252

[15] Taylor, Joseph L. Functions of several noncommuting variables Bull. Amer. Math. Soc. 1973 1 34

[16] Taylor, Michael E. Noncommutative harmonic analysis 1986

[17] Vasilescu, Florian-Horia Analytic functional calculus and spectral decompositions 1982