Geometric Properties of the Pentablock
- PDF / 319,327 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 235 Views
Complex Analysis and Operator Theory
Geometric Properties of the Pentablock Guicong Su1 Received: 23 January 2020 / Accepted: 13 April 2020 © The Author(s) 2020
Abstract In this paper, we give a positive answer to the question raised in Kosi´nski (Complex Anal Oper Theory 9(6):1349–1359, 2015) and Zapałowski (J Math Anal Appl 430(1):126–143, 2015), i.e., we show that the pentablock P is a C-convex domain.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Pentablock . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pentablock as a Hartogs Domain . . . . . . . . . . . . . . . 2.3 Some Useful Results . . . . . . . . . . . . . . . . . . . . . 2.4 C-convex Domain . . . . . . . . . . . . . . . . . . . . . . . 3 The Set of All Tangent Hyperplanes to P at the Non-smooth Part 4 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
1 Introduction Recently, many authors showed great interest in two domains: the symmetrized bidisc and the tetrablock, arising from the μ-synthesis, from the aspect of geometric function theory. Actually, both domains are C-convex but non-convex, and they cannot be exhausted by domains biholomorphic to convex ones, with the Lempert’s theorem (see Lempert [13,14]) holding on these two domains, i.e., the Lempert function and the Carathéodory distance coincide on them (see [2,6–8,20]). So from the point of view of the Lempert’s theorem holding, these two domains play an important role in
Communicated by Filippo Bracci. This work is partially supported by NCN Grant SONATA BIS No. 2017/26/E/ST1/00723.1.
B 1
Guicong Su [email protected] Instytut Matematyki, Uniwersytet Jagiello´nski, Łojasiewicza 6, 30-348 Kraków, Poland 0123456789().: V,-vol
44
Page 2 of 14
G. Su
the study of a long-standing open problem whether Lempert’s theorem still holds for C-convex domain. However, as far as we know, the answer is positive for C-convex domain with C 2 boundary (see [10]). In 2015, Agler, Lykova and Young [1] introduced a new bounded domain P by 2 P := (a21 , tr A, det A) : A = [ai j ]i, j=1 ∈ B , where B := A ∈ C2×2 : ||A|| < 1 denotes the open unit ball in the space C2×2 with the usual operator norm. They called this domain the pentablock as P ∩ R3 is a convex body bounded by five faces, three of which are flat and two are curved (see [1]). The pentablock P is polynomially convex and starlike about the origin, but neither circled nor convex. Moreover, it does not have a C 1 boundary (see [1]). This new domain is also arising from the μ-synth
Data Loading...
