scholarly journals REGULAR FUNCTIONS ON THE SHILOV BOUNDARY

2005 ◽  
Vol 04 (06) ◽  
pp. 613-629 ◽  
Author(s):  
OLGA BERSHTEIN
Keyword(s):  
Symmetric Domain ◽  
Principal Series ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Shilov Boundary ◽  
Generators And Relations ◽  
Regular Functions ◽  
Degenerate Principal Series

In this paper a *-algebra of regular functions on the Shilov boundary S(𝔻) of bounded symmetric domain 𝔻 is constructed. The algebras of regular functions on S(𝔻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(𝔻) = Un is investigated.

scholarly journals Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

2015 ◽  
Vol 117 (2) ◽  
pp. 203 ◽  
Author(s):  
P. Mellon
Keyword(s):  
Fixed Point ◽  
Boundary Component ◽  
Symmetric Domain ◽  
Bounded Symmetric Domain ◽  
Bounded Symmetric Domains ◽  
Linear Isometry ◽  
Holomorphic Map ◽  
Infinite Dimensional ◽  
Hilbert Ball

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.

scholarly journals Orbits of triples in the Shilov boundary of a bounded symmetric domain

2006 ◽  
Vol 11 (3) ◽  
pp. 387-426 ◽  
Author(s):  
Jean-Louis Clerc ◽  
Karl-Hermann Neeb
Keyword(s):  
Symmetric Domain ◽  
Bounded Symmetric Domain ◽  
Shilov Boundary

THE N-WIDTHS OF HARDY–SOBOLEV AND BERGMAN–SOBOLEV SPACES OF COMPLEX VARIABLES

2004 ◽  
Vol 02 (04) ◽  
pp. 309-335
Author(s):  
HONGMING DING
Keyword(s):  
Unit Ball ◽  
Sobolev Spaces ◽  
Unit Sphere ◽  
Symmetric Domain ◽  
Complex Variables ◽  
Linear Operators ◽  
Bounded Symmetric Domain ◽  
Asymptotic Estimates ◽  
Shilov Boundary ◽  
Optimal Linear

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For R≥1, l∈ℤ+ and 1≤p≤∞, let DR=RD, Hp,l(DR) and Ap,l(DR) be the Hardy–Sobolev and Bergman–Sobolev spaces on DR, respectively. In this paper we show that the Kolmogorov, linear, Gel'fand, and Bernstein N-widths of Hp,l(DR) in Lp(Σ) all coincide, calculate the exact value, and identify optimal subspaces or optimal linear operators. We also do the same for N-widths of Ap,l(DR) in Lp(D). Moreover, we obtain new asymptotic estimates for the linear and Gel'fand N-widths of [Formula: see text] and [Formula: see text] in Lq(Sn) and Lq(Bn), where R>1, l∈ℤ+, 2≤p≤q≤∞, [Formula: see text], [Formula: see text] are the unit ball and unit sphere in [Formula: see text], respectively, and [Formula: see text]. Furthermore, we obtain asymptotic estimates for the linear and Gel'fand N-widths of Hp,l(DR) in Lq(Σ), where R>1, l∈ℤ+ and 2≤p≤q≤∞.

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