Gindikin, S.G. \Analysis on homogeneous domains". Russian Math. Surveys 19 (4) 1964, pp. 1-89.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the spherical function (x) on the real bounded symmetric domain DR is given by (x) i = i (x) x 2 DR : 2.6. The function on symmetric cones. The H invariant measure on is given by d 0 = x) where d = dimR (J) dim C (V ) The Gindikin Koecher Gamma function [11] associated with the convex, symmetric cone is de ned by ( x) x) d=r The integral converges if and only if Re( j ) j 1)a=2 for j = 1; 2; r, 10] Theorem VII.1.1. Using the identi cation a via the map ( 1 ; n ) 1 1 r r we have ....

S. G. Gindikin, Analysis on homogeneous domains, Uspekhi Mat. Nauk, 19, 1964, 3|92; Russian Math. Surveys, 19, no. 4, 1-89

....DA CB C D A = 1: The cone in J = Herm(n) can be identi ed as the set of positive de nite Hermitian matrices. The group SU(n; n) acts on T( iJ by linear fractional transformations: T = AT B) CT D) be the Gindikin Koecher Gamma function associated with the cone see [3, 4]: tr(x) det(x) dx : Assume that 2n 1. Let H (T( 9 be the space of holomorphic functions F : T( kFk : T( jF (x iy)j det(y) 2n dxdy 1 (11) n (4 ) n) Then H (T( 7 is a Hilbert space and the group SU(n; n) acts unitarily and ....

S. G. Gindikin, Analysis on homogeneous domains, Uspekhi Mat. Nauk, 19, No 4, 3-92; Russian Math. Surveys, 19, No. 4, 1-89.

....which we try to choose as weak as possible. Our proof for atomic decomposition is new, and can be generalized to all tube domains over homogeneous cones. 2. The forward light cone 2.1. Group action. The forward light cone is a symmetric cone. Such cones have been studied by Gindikin in [G]. Nowadays, the book of Faraut and Kor anyi [FK] is a very good reference for their study. There, the cone is called the Lorentz cone. When we say that it is a symmetric cone, we mean that it satis es the two properties it is self dual, which means that, for x 2 R , the scalar product x:y ....

S.G. Gindikin, Analysis on homogeneous domains, Russian Math. Surveys 19 (1964), 1-89.

....simply transitively on the cone. That is, Z f(h Gamma1 y) d(y) h) Z f(y) d(y) 8 f 2 L 1 (d) h 2 H; where is a character of the group H. Such measures appear in different contexts related to symmetric cones and Siegel domains, being completely characterized by Gindikin in [6] [5] (see also x2:3 below) The particular choice = ffi 0 (the delta distribution at the origin) corresponds to the classical Hardy space on the tube: H p (T Omega ) n F 2 H(T Omega ) kFkH p = sup y2 Omega Z R n jF (x iy)j p dx 1 p 1 o : On the other hand, the Lebesgue ....

....Further, there is subgroup H j of H and a point t j 2 Omega , such that Omega j = H j t j . 4. Given 2 Xi, then co (Supp ) Omega if and only if 1 6= 0. We point out that the preceding result is not completely new, since parts of it are contained in earlier works of Gindikin (see [6] [5]) However, for the sake of completeness we shall present here a more affordable proof, using the modern notation in the text [3] The paper will be structured as follows. In x2 we recall the basic notions of symmetric cones and quasi invariant measures. In x3 we study the spaces H p (T Omega ....

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Gindikin, S.G. "Analysis on homogeneous domains". Russian Math. Surveys 19 (4) 1964, pp. 1-89.

....(Fock) space on C n ,theBergman projector on the space of holomorphic functions on the unit ball B n in C n and the Cauchy Szego projector on the Hardy space. The representation of the latter projector as a convolution operator on the Heisenberg group was obtained by Gindikin in [8], but representations of the other two projectors as convolutions seem to be new. Now we would like to give the basic notions, notations and references, which will used in Section 3. The Heisenberg group H n (see [18, Chap. 1] or [17, Chap. XII] is a step 2 nilpotent Lie group. As a C # ....

....[15, 2.2.2(v) It immediately follows from the comparison of (3.11) and (3.12) that f(#) # B f(#) 1 ##,##) n 1 d#(#) The last formula is the integral representation with the Bergman kernel for square integrable holomorphic functions on the unit ball in C n . # Corollary 3. 4 [8] The orthogonal projector Szego on the boundary bU n of the upper half space in C n 1 has the kernel S(z,w) i 2 (w n 1 z n 1 ) n # j=1 z j w j ) n 1 . Proof. It is well known [8, 9, 17] that there is a unitary representation of the Heisenberg group H n as the simply ....

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S.G. Gindikin, Analysis on homogeneous domains, Russian Math. Surveys 19(4), 1964, 1--89.

....are under a subgroup H which acts simply transitively on In the above range of , the measures are locally nite with full support in and the corresponding spaces H p (T ) are of Bergman type. These general spaces have been previously considered in the literature by di erent authors: [5], 3] However, the Hardy type spaces obtained from the singular measures in the analytic continuation of (1.3) seem not to have been studied before, even when p = 2. The purpose of this paper is precisely to collect some of the L p properties that all these spaces H p (T ) share with the ....

.... L 2 s = L 2( s (2 )d ) L 2( s ( 2 ) 1 )d ) The following theorem characterizes the whole family of spaces H 2 (T ) We recall that this type of characterization, for some of the spaces mentioned in the introduction, has been previously given by di erent authors [5], 7] 3] For completeness, we include a simple proof that covers all the family of spaces. THEOREM 4.1 For every F 2 H 2 (T ) there exists f 2 L 2 s ( such that F (z) 1 (2 ) n 2 Z e i(zj ) f( s (2 ) d ; z 2 T : 4.2) Conversely, if f 2 L 2 s then the ....

Gindikin, S.G. \Analysis on homogeneous domains". Russian Math. Surveys 19 (4) 1964, pp. 1-89.

....X k=1 t kk A k X m k T mk (t kk 2 R; T mk 2 h mk ) 2.1) For the element T 2 h in (2.1) we put L k : X m k T mk (1 k r 0 1) 2.2) 8 Given T 2 h, the symbols T mk (m k) as well as L k will be used to denote the elements in (2.1) and (2.2) without any comments in this paper. Let 5 be the open subset of h defined by 5 : f T 2 h j t kk 0 for all k = 1; 2; r g : Putting T kk : 2 log t kk )A k (1 k r) for T 2 5, we set fl(T ) exp T 11 1 exp L 1 1 exp T 22 1 1 1 exp L r01 1 exp T rr : 2.3) Then fl(T ) 2 H, and we shall write it in the following form of lower ....

....0 rr 1 C C A = 0 B B t 00 11 T 00 21 t 00 22 . T 00 r1 T 00 r2 t 00 rr 1 C C A with t 00 kk = t kk t 0 kk (1 k r) 2.5) T 00 mk = t mmT 0 mk X k l m [T ml ; T 0 lk ] t 0 kk T mk (1 k m r) 2. 6) ii) The map fl is a diffeomorphism from 5 onto H. Proof. i) Let 1 i k r. We have exp L k 1 exp L 0 i = exp Ad(exp L k )L 0 i 1 exp L k = exp e ad L k L 0 i 1 exp L k : 2.7) In the same way, we get exp T kk 1 exp L 0 i = exp e adT kk L 0 i 1 exp T kk ; 2.8) exp L k 1 exp T 0 ii = exp T 0 ii 1 exp L k (i k) ....

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S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys, 19 (1964), 1--89.

....a hyperbolic polynomial, that is, whether it is possible to express K 0 as the hyperbolicity cone of a hyperbolic polynomial. In this section, we show that this is indeed the case. Our results depend heavily on the classification of homogeneous cones by Vinberg [31] and the results of Gindikin [10, 11]. Here we follow Gindikin [11] exclusively. Our presentation will be brief; in addition to the above references, some introductory information on homogeneous cones can be found in Guler [12] and more detailed exposition on closely related topics will be given in the forthcoming paper Guler and ....

Gindikin, S. G. (1964). Analysis in homogeneous domains. Russian Math. Surveys 19 1--89.

....is the first to use these properties to obtain an algebraic classification of these cones. Siegel domains described below play an essential role in the classification and in the description of algebraic construction of homogeneous cones. The literature on these two topics is large, see for example [9, 2, 3, 8, 1], etc. Here we describe only the basic elements of this theory and those aspects of it that we need in order to calculate self concordant barrier functions and barrier parameters # for homogeneous cones. Definition 3.1 Let K be a cone in R k . A K bilinear symmetric form B(u; v) in R p is a ....

....affine homogeneous. This can be seen by checking that the following affine transformations form a transitive subgroup of S(K; B) A 1 (x; u) x 2B(u; a) B(a; a) u a) a 2 R p ; A 2 (x; u) gx; gu) g 2 G Aut(K) The following remarkable characterization theorem is due to Vinberg, see [9, 2]. Theorem 3.1 Any affine homogeneous domain D R n is affine equivalent to a Siegel domain. The cone fitted to a Siegel domain S(K; B) which we call the Siegel cone of K and B, is given by SC(K;B) f(x; u; t) 2 R k Theta R p Theta R : t 0; tx Gamma B(u; u) 2 Kg: We remark that a ....

S. G. Gindikin, Analysis in homogeneous domains. Russian Math. Surveys 19 (1964), pp. 1--89.

....where Q : L 2 (H n ) L 2 (H n ) is a pseudodifferential operator. Obviously TQ : H 2 (H n ) H 2 (H n ) The invariance of the tangential Cauchy Riemann equations under right shifts of H n implies that the Szego projector can be realized as a (left) convolution operator on H n [23] (see also Corollary 5.14) Thus the algebra of the Toeplitz operators on H n can be naturally imbedded into the algebra of (pseudodifferential) operators generated by left group convolutions on H n and PDO. First, let us consider a case of a pre symbol Q taken from usual Euclidean ....

....of B is an involution [46, 2.2.2(v) It immediately follows from the comparison of (49) and (50) that: f(AE) Z B f(i) 1 Gamma hAE; ii) n 1 d(i) The last formula is the integral representation with the Bergman kernel for holomorphic functions on unit ball in C n . Corollary 5. 14 [23] The orthogonal projector Szego on the boundary bU n of the upper half space in C n 1 has the kernel S(z; w) i 2 ( w n 1 Gamma z n 1 ) Gamma n X j=1 z j w j ) Gamman Gamma1 : Proof. It is well known [23, 24, 49] and was described at Example 4.2, that there is a unitary ....

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Simon G. Gindikin. Analysis on homogeneous domains. Russian Math. Surveys, 19(4):1--89, 1964.

....a hyperbolic polynomial, that is, whether it is possible to express K 0 as the hyperbolicity cone of a hyperbolic polynomial. In this section, we show that this is indeed the case. Our results depend heavily on the classification of homogeneous cones by Vinberg [26] and the results of Gindikin [8, 9]. Here we follow Gindikin [9] exclusively. Our presentation will be brief; in addition to the above references, some introductory information on homogeneous cones can be found in Guler [10] and more detailed exposition on closely related topics will be given in the forthcoming paper Guler and ....

Gindikin, S. G. (1964). Analysis in homogeneous domains. Russian Math. Surveys 19 1--89.

....is the first to use these properties to obtain an algebraic classification of these cones. Siegel domains described below play an essential role in the classification and in the description of algebraic construction of homogeneous cones. The literature on these two topics is large, see for example [11, 3, 4, 10, 1], etc. Here we describe only the basic elements of this theory and those aspects of it that we need in order to calculate self concordant barrier functions and barrier parameters # for homogeneous cones. Definition 3.1 Let K be a cone in R k . A K bilinear symmetric form B(u; v) in R p is a ....

.... can be seen by checking that the following affine transformations form a transitive subset of S(K; B) A 1 (x; u) x 2B(u; a) B(a; a) u a) a 2 R p ; A 2 (x; u) gx; gu) g 2 G Aut(K) BARRIER PARAMETER 5 The following remarkable characterization theorem is due to Vinberg, see [11, 3]. Theorem 3.1 Any affine homogeneous domain D R n is affine equivalent to a Siegel domain. The cone fitted to a Siegel domain S(K; B) which we call the Siegel cone of K and B, is given by SC(K;B) f(x; u; t) 2 R k Theta R p Theta R : t 0; tx Gamma B(u; u) 2 Kg: We remark that a more ....

S. G. Gindikin, Analysis in homogeneous domains. Russian Math. Surveys 19 (1964), pp. 1--89.

....much larger than the class of homogeneous self dual cones. However, homogeneous cones can also be classified in terms of a class of non associative matrix algebras, called T algebras by Vinberg, see Vinberg [32, 33] These cones can be constructed recursively, see Vinberg [32, 33] Gindikin [10], Rothaus [27] Dorfmeister [4, 5, 6] etc. The Hessian of the characteristic function defines an invariant Riemannian metric in K. Thus, the characteristic function has intimate connections with Lie groups and differential geometry [18, 19, 26, 32, 4, 5, 28] The characteristic function also ....

....The Hessian of the characteristic function defines an invariant Riemannian metric in K. Thus, the characteristic function has intimate connections with Lie groups and differential geometry [18, 19, 26, 32, 4, 5, 28] The characteristic function also has uses in carrying out Fourier analysis on K [15, 10, 30, 8]. The paper is organized as follows. In Section 2 we present some concepts and results from the theory of convex cones, especially concepts related to the automorphism group of the cone. In Section 3, we introduce the characteristic function of a cone K R n and discuss its invariance ....

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Gindikin, S. G. (1964) Analysis in homogeneous domains. Russian Math. Surveys 19 1--89.

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Gindikin, S.G. \Analysis on homogeneous domains". Russian Math. Surveys 19 (4) 1964, pp. 1-89.

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S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys 19 (1964), 1--89.

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