Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 810735, 7 pages
http://dx.doi.org/10.1155/2013/810735
Research Article
Hardy-Type Space Associated with an Infinite-Dimensional
Unitary Matrix Group
Oleh Lopushansky
Institute of Mathematics, University of Rzeszów, 16A Rejtana Street, 35-310 Rzeszów, Poland
Correspondence should be addressed to Oleh Lopushansky; [email protected]
Received 23 March 2013; Accepted 18 June 2013
Academic Editor: Qing-Wen Wang
Copyright © 2013 Oleh Lopushansky. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which
is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding
Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space
and an associated symmetric Fock space is established.
1. Introduction
2. Preliminaries
We investigate an orthogonal system of the Hilbert-Schmidt
polynomials in the space 𝐿2𝜒 of square-integrable complex
functions on the projective limit U = lim𝑈(𝑚) of unitary
←
(𝑚 × 𝑚)-dimensional matrix groups 𝑈(𝑚) (𝑚 ∈ N), called
the space of virtual unitary matrices and endowed with the
projective limit measure 𝜒 = lim𝜒𝑚 of the probability Haar
←
measures 𝜒𝑚 on 𝑈(𝑚). The measure 𝜒 on the space U is
invariant under the right action of the infinite-dimensional
unitary group 𝑈(∞) × 𝑈(∞), where 𝑈(∞) = ⋃𝑚 𝑈(𝑚).
The space of virtual unitary matrices U was studied by
Neretin [1] and Olshanski [2]. This notion relates to D.
Pickrell’s space of virtual Grassmannian [3] and to Kerov,
Olshanski, and Vershik’s space of virtual permutations [4].
Various spaces of integrable functions with respect to measures that are invariant under infinite-dimensional groups
have been widely applied in stochastic processes [5], infinitedimensional probability [6, 7], complex analysis [8], and so
forth.
The main results of the present paper are Theorems 67 that describe a Hardy-type subspace H2𝜒 ⊂ 𝐿2𝜒 spanned
by the finite type homogenous Hilbert-Schmidt polynomials
that are generated by an associated symmetric Fock space.
We consider the following infinite-dimensional unitary
matrix groups:
𝑈 (∞) = ⋃ {𝑈 (𝑚) : 𝑚 ∈ N} ,
𝑈2 (∞) := 𝑈 (∞) 𝑈 (∞) ,
(1)
where 𝑈(𝑚) is the group of unitary (𝑚 × 𝑚)-matrices which
is identified with the subgroup in 𝑈(𝑚 + 1) fixing the (𝑚 +
1)th basis vector. In other words, 𝑈(∞) is the group of
infinite unitary matrices 𝑢 = [𝑢𝑖𝑗 ]𝑖,𝑗∈N with finitely many
matrix entries 𝑢𝑖𝑗 distinct from 𝛿𝑖𝑗 . We equip every group
𝑈(𝑚) with the probability Haar measure 𝜒𝑚 .
Following [1, 2], every matrix 𝑢𝑚 ∈ 𝑈(𝑚) with 𝑚 > 1, we
write in the following block matrix form:
𝑢𝑚 = [
𝑧𝑚−1 𝑎
],
𝑏 𝑡
(2)
corresponding to the partition 𝑚 = (𝑚 − 1) + 1 so that 𝑧𝑚−1 ∈
𝑈(𝑚 − 1) and 𝑡 ∈ C. Over the group 𝑈(∞) (resp., 𝑈(𝑚)) the
right action is well defined:
𝑢 ⋅ 𝑔 = 𝑤−1 𝑢V,
(3)
2
Abstract and Applied Analysis
where 𝑢 belongs to 𝑈(∞) (resp., to 𝑈(𝑚)) and 𝑔 = (V, 𝑤)
belongs to 𝑈2 (∞) (resp., to 𝑈2 (𝑚) := 𝑈(𝑚) × 𝑈(𝑚)). In
[1, Proposition 0.1], [2, Lemma 3.1], it was proven that the
following Livšic-type mapping:
𝑚
: 𝑈 (𝑚) ∋ 𝑢𝑚 󳨀→ 𝑢𝑚−1 ∈ 𝑈 (𝑚 − 1) ,
𝜋𝑚−1
(4)
such that
[
−1
− 𝑎(1 + 𝑡) 𝑏 : 𝑡 ≠ − 1,
𝑧
𝑧𝑚−1 𝑎
] 󳨃󳨀→ { 𝑚−1
𝑏 𝑡
𝑧𝑚−1 :
𝑡 = −1,
(5)
(which is not a group homomorphism) is Borel and surjective
onto 𝑈(𝑚−1) and commutes with the right action of 𝑈2 (𝑚−
1).
As is known [1, Theorem 1.6], the pullback of the
probability Haar measure 𝜒𝑚−1 on 𝑈(𝑚 − 1) under the
𝑚
is the probability Haar measure 𝜒𝑚 on 𝑈(𝑚),
mapping 𝜋𝑚−1
that is,
𝜒𝑚−1 ∘
𝑚
𝜋𝑚−1
= 𝜒𝑚 .
(6)
Let 𝑈󸀠 (𝑚) ⊂ 𝑈(𝑚) be the subset of unitary matrices
which do not have {−1}, as an eigenvalue. Then, 𝑈󸀠 (𝑚) is
open in 𝑈(𝑚), and the complement 𝑈(𝑚) \ 𝑈󸀠 (𝑚) is a 𝜒𝑚 negligible set. Moreover (see [2, Lemma 3.11]), the mapping
𝑚
𝜋𝑚−1
󸀠
󸀠
: 𝑈 (𝑚) 󳨀→ 𝑈 (𝑚 − 1)
(7)
is continuous and surjective.
Consider the projective limits, taken with respect to
𝑚
and their continuous
the surjective Borel projections 𝜋𝑚−1
𝑚
restrictions 𝜋𝑚−1 |𝑈󸀠 (𝑚) , respectively,
U󸀠 = lim 𝑈󸀠 (𝑚) ,
U = lim 𝑈 (𝑚) ,
←󳨀
←󳨀
(8)
called the spaces of virtual unitary matrices. Notice that U is
a Borel subset in the Cartesian product ⨉𝑚∈N 𝑈(𝑚) = {𝑢 =
(𝑢𝑚 ) : 𝑢𝑚 ∈ 𝑈(𝑚)} endowed with the product topology,
𝑚
are Borel. Moreover, the canonical
because all mapping 𝜋𝑚−1
projections
𝜋𝑚 : U󸀠 󳨀→ 𝑈󸀠 (𝑚) ,
𝜋𝑚 : U 󳨀→ 𝑈 (𝑚) ,
(9)
𝑚
∘ 𝜋𝑚 , are surjective by surjectivity
such that 𝜋𝑚−1 = 𝜋𝑚−1
𝑚
𝑚
of 𝜋𝑚−1 and 𝜋𝑚−1 |𝑈󸀠 (𝑚) .
Following [2, Lemma 4.8], [1, Section 3.1], with the help
of the Kolmogorov consistent theorem, we uniquely define
a probability measure 𝜒 on U󸀠 as the projective limit under
the mapping (6),
𝜒 = lim𝜒𝑚 ,
(10)
←󳨀
which satisfies the equality 𝜒 = 𝜒𝑚 ∘𝜋𝑚 for all 𝑚 ∈ N. On U\
U󸀠 , the measure 𝜒 is zero, because 𝜒𝑚 is zero on 𝑈(𝑚) \
𝑈󸀠 (𝑚) for all 𝑚 ∈ N.
Using (3), right action of the group 𝑈2 (∞) on the space
of virtual unitary matrices U can be defined (see [2, Definition 4.5]) as follows:
𝜋𝑚 (𝑢 ⋅ 𝑔) = 𝑤−1 𝜋𝑚 (𝑢) V,
𝑢 ∈ U,
where 𝑚 is so large that 𝑔 = (V, 𝑤) ∈ 𝑈2 (𝑚).
(11)
The canonical dense embedding 𝚤 : 𝑈(∞) 󴀜󴀠 U to any
element 𝑢𝑚 ∈ 𝑈(𝑚) assigns the unique sequence 𝑢 = (𝑢𝑙 )𝑙∈N ,
such that
𝚤 : 𝑈 (𝑚) ∋ 𝑢𝑚 󳨃󳨀→ (𝑢𝑙 ) ∈ U,
𝑚
(𝑢𝑚 ) : 𝑙 < 𝑚,
𝜋𝑙𝑙+1 ∘ ⋅ ⋅ ⋅ ∘ 𝜋𝑚−1
{
{
{
{𝑢𝑚 :
𝑙 = 𝑚,
𝑢𝑙 = {
{
𝑢
0
{[ 𝑚
{
]:
𝑙 > 𝑚,
{ 0 1𝑙−𝑚
(12)
where 1𝑙−𝑚 is the unit in 𝑈(𝑙 − 𝑚). So, the image 𝚤 ∘
𝑈(∞) consists of stabilizing sequences in U (see [2, Section
4]).
3. Invariant Probability Measure
In what follows, we will endow the space of virtual unitary
matrices U with the measure 𝜒 = lim𝜒𝑚 . A complex func←
tion on U is called cylindrical [2, Definition 4.5] if it has the
following form:
𝑓 (𝑢) = (𝑓𝑚 ∘ 𝜋𝑚 ) (𝑢) ,
𝑢 ∈ U,
(13)
for a certain 𝑚 ∈ N and a certain complex function 𝑓𝑚 on
𝑈(𝑚).
Any continuous bounded function 𝑓 on U󸀠 has a unique
𝜒-essentially bounded extension on U, because the set U \ U󸀠
is 𝜒-negligible. Therefore, if the function 𝑈󸀠 (𝑚) ∋ 𝜋𝑚 (𝑢) 󳨃→
𝑓𝑚 [𝜋𝑚 (𝑢)] in the definition (13) is continuous and bounded,
then the corresponding cylindrical function 𝑓 is 𝜒 essentially
bounded.
By L∞
𝜒 , we denote closure of the algebraic hull of all
cylindrical 𝜒-essentially bounded functions (13) with respect
to the following norm:
󵄨
󵄩󵄩 󵄩󵄩
󵄨
= ess sup 󵄨󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 .
󵄩󵄩𝑓󵄩󵄩L∞
(14)
𝜒
𝑢∈U
Lemma 1. The measure 𝜒 = lim𝜒𝑚 on U is a Radon proba←
bility measure such that
∫ 𝑓 (𝑢 ⋅ 𝑔) 𝑑𝜒 (𝑢) = ∫ 𝑓 (𝑢) 𝑑𝜒 (𝑢) ,
U
U
(15)
for all 𝑔 ∈ 𝑈2 (∞) and 𝑓 ∈ L∞
𝜒 . For any compact set 𝐾 ⊂
𝑈(𝑚) the following equality holds:
(𝜒 ∘ 𝚤) (𝐾) = 𝜒𝑚 (𝐾) .
(16)
Proof. Recall the Prohorov criterion, which is adapted to our
notation (see [9, Chapter IX.4.2, Theorem 1] or [6, Theorem
6]): there exists a Radon probability measure 𝜒󸀠 on U󸀠 such
that
󵄨
𝜒󸀠 = 𝜒𝑚 ∘ 𝜋𝑚 󵄨󵄨󵄨U󸀠
∀𝑚 ∈ N,
(17)
if and only if for every 𝜀 > 0 there exists a compact set K in
U󸀠 such that the following inequality
(𝜒𝑚 ∘ 𝜋𝑚 ) (K) ≥ 1 − 𝜀
∀𝑚 ∈ N
(18)
Abstract and Applied Analysis
3
holds; in this case, 𝜒󸀠 is uniquely determined by means of
the formula 𝜒󸀠 (K) = inf 𝑚∈N (𝜒𝑚 ∘ 𝜋𝑚 )(K), where K is a
compact set in U󸀠 .
Let 𝐾𝑛 ⊂ 𝑈󸀠 (𝑛) be a compact set with a fixed 𝑛. Putting
𝑛
(𝐾𝑛 ), we have
𝐾𝑛−1 = 𝜋𝑛−1
𝑛
𝜒𝑛−1 (𝐾𝑛−1 ) = (𝜒𝑛−1 ∘ 𝜋𝑛−1
) (𝐾𝑛 ) = 𝜒𝑛 (𝐾𝑛 ) .
(19)
is generated by the right action of 𝑈2 (∞) over U. Choosing
instead of 𝑈(∞) a compact subgroup 𝑈(𝑚) or the compact
subgroups
𝑈0 = {𝑔0 (𝜗) = exp (i𝜗) : 𝜗 ∈ (−𝜋, 𝜋]} ,
𝑈𝑗 (𝑚) = {𝑔𝑚𝑗 (𝜗) = 1𝑗−1 ⊗ exp (i𝜗) ⊗ 1𝑚−𝑗 : 𝜗 ∈ (−𝜋, 𝜋]}
𝑗 = 1, . . . , 𝑚,
(26)
On the other hand, if we put 𝐾𝑛+1 = [ 𝐾0𝑛 01 ], then via (6),
𝜒𝑛+1 (𝐾𝑛+1 ) = (𝜒𝑛 ∘ 𝜋𝑛𝑛+1 ) (𝐾𝑛+1 )
= (𝜒𝑛 ∘ 𝜋𝑛𝑛+1 ) [
(20)
𝐾𝑛 0
] = 𝜒𝑛 (𝐾𝑛 ) .
0 1
As a consequence, the compact set K = (𝐾𝑚 ) in U󸀠 ,
generated by a compact set 𝐾𝑛 ⊂ 𝑈󸀠 (𝑛) with the help of
𝑛
, satisfies the following condition:
mappings 𝜋𝑛−1
𝜒𝑛 (𝐾𝑛 ) = 𝜒𝑚 (𝐾𝑚 )
∀𝑚 ∈ N.
Lemma 2. For any 𝑓 ∈ L∞
𝜒 the following equalities:
∫ 𝑓𝑑𝜒 = ∫ 𝑑𝜒 (𝑢) ∫
U
(21)
The probability Haar measure 𝜒𝑛 is regular on 𝑈(𝑛), and
the complement 𝑈(𝑛) \ 𝑈󸀠 (𝑛) is a negligible set. Hence,
if 𝐾𝑛 runs over all compact sets in 𝑈󸀠 (𝑛), then
sup 𝜒𝑛 (𝐾𝑛 ) = 1.
(22)
𝐾𝑛 ⊂𝑈󸀠 (𝑛)
Therefore, for every 𝜀 > 0 there exists a compact set 𝐾𝑛 ⊂
𝑈󸀠 (𝑛) such that 𝜒𝑛 (𝐾𝑛 ) ≥ 1 − 𝜀. From (21), it follows that
for every 𝜀 > 0 the compact set K satisfies the hypothesis
of Prohorov’s criterion:
(𝜒𝑚 ∘ 𝜋𝑚 ) (K) = 𝜒𝑚 (𝐾𝑚 ) ≥ 1 − 𝜀
we obtain the corresponding subgroups of shifts 𝑄𝑔 with elements 𝑔 ∈ 𝑈2 (𝑚) or with elements 𝑔0 (𝜗) ∈ 𝑈02 and 𝑔𝑚𝑗 (𝜗) ∈
𝑈𝑗2 (𝑚), respectively. Here, 1𝑚 means the unit element in
𝑈(𝑚).
∀𝑚 ∈ N.
(23)
So, in view of this criterion, there exists a unique Radon probability measure 𝜒󸀠 on U󸀠 which satisfies the condition (17).
However, on the projective limits U󸀠 = lim𝑈󸀠 (𝑚), there exists
𝑈2 (𝑚)
U
∫ 𝑓 𝑑𝜒 =
U
𝑄𝑔 𝑓 (𝑢) 𝑑 (𝜒𝑚 ⊗ 𝜒𝑚 ) (𝑔) , (27)
𝜋
1
∫ 𝑑𝜒 (𝑢) ∫ 𝑄𝑔(𝜗) 𝑓 (𝑢) 𝑑𝜗,
2𝜋 U
−𝜋
(28)
with 𝑔(𝜗) ∈ 𝑈02 or 𝑈𝑗2 (𝑚) hold.
Proof. For any 𝑓 ∈ L∞
𝜒 , the function (𝑢, 𝑔) 󳨃→ 𝑄𝑔 𝑓(𝑢) =
𝑓(𝑢 ⋅ 𝑔) is integrable on the Cartesian product U × 𝑈2 (𝑚). By
the Fubini theorem, we obtain
∫ 𝑑𝜒 (𝑢) ∫
𝑈2 (𝑚)
U
=∫
𝑈2 (𝑚)
𝑄𝑔 𝑓 (𝑢) 𝑑 (𝜒𝑚 ⊗ 𝜒𝑚 ) (𝑔)
(29)
𝑑 (𝜒𝑚 ⊗ 𝜒𝑚 ) (𝑔) ∫ 𝑄𝑔 𝑓 (𝑢) 𝑑𝜒 (𝑢) .
U
This equality yields the required formula (27), because
the internal integral on the right-hand side is independent
of 𝑔 and ∫𝑈2 (𝑚) 𝑑(𝜒𝑚 ⊗ 𝜒𝑚 ) = 1. In turn, putting instead
of 𝑈(𝑚) the subgroups 𝑈0 and 𝑈𝑗 (𝑚), we obtain equalities
(28).
←
a unique 𝑈2 (∞)-invariant Radon measure 𝜒, determined by
the equality (15). Using the uniqueness property of projective
limits, we obtain 𝜒󸀠 = 𝜒. The measure 𝜒 on U\U󸀠 is defined
to be zero, because 𝜒𝑚 is zero on 𝑈(𝑚) \ 𝑈󸀠 (𝑚).
As a consequence of (21), we obtain (16), because
𝜒 (K) = inf 𝜒𝑚 (𝐾𝑚 ) = 𝜒𝑛 (𝐾𝑛 ) .
(24)
𝑚∈N
2
As is known [1, Proposition 3.2], the measure 𝜒 is 𝑈 (∞)invariant under the right actions (11) on the space U. Hence,
for every 𝑓 ∈ L∞
𝜒 , the equality (15) holds.
4. Shift Groups
𝑄𝑔 𝑓 (𝑢) = 𝑓 (𝑢 ⋅ 𝑔) ,
𝑔 ∈ 𝑈 (∞) 𝑢 ∈ U,
Consider the countable orthogonal Hilbertian sum
E := ⨁C𝑚 = {𝑥 = (𝑥𝑚 ) : 𝑥𝑚 ∈ C𝑚 , ‖𝑥‖E < ∞} ,
𝑚∈N
(30)
with the scalar product ⟨𝑥 | 𝑦⟩E = ∑𝑚 ⟨𝑥𝑚 | 𝑦𝑚 ⟩C𝑚 , where
every coordinate 𝑥𝑚 ∈ C𝑚 is identified with its image (0, . . . ,
0, 𝑥𝑚 , 0, . . .) ∈ E under the embedding C𝑚 󴀜󴀠 E.
Let ⊗𝑛h E stand for the complete 𝑛th tensor power of
the Hilbert subspace E, endowed with the Hilbertian scalar
product and norm, respectively,
⟨𝑥1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑥𝑛 | 𝜓𝑛 ⟩⊗𝑛 E = ∑⟨𝑥1 | 𝑦1𝑗 ⟩E . . . ⟨𝑥𝑛 | 𝑦𝑛𝑗 ⟩E ,
Consider that in the space L∞
𝜒 , the group of shifts
2
5. The Homogeneous Hilbert-Schmidt
Polynomials
h
(25)
𝑗
1/2
󵄩󵄩 󵄩󵄩
󵄩󵄩𝜓𝑛 󵄩󵄩⊗𝑛 E = ⟨𝜓𝑛 | 𝜓𝑛 ⟩⊗𝑛 E ,
h
h
(31)
4
Abstract and Applied Analysis
where 𝑥1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑥𝑛 , 𝑦1𝑗 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑦𝑛𝑗 ∈ ⊗𝑛h E with 𝑥𝑡𝑗 , 𝑦𝑡𝑗 ∈ E for
all 𝑡 = 1, . . . , 𝑛 and 𝜓𝑛 = ∑𝑗 𝑦1𝑗 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑦𝑛𝑗 denotes a finite
sum. Put ⊗0h E = C. We use the following short denotation:
𝑥⊗𝑛 = 𝑥 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑥,
𝑥 ∈ E.
(32)
𝑚
Replacing the space E by the subspace C , we similarly
define the tensor product ⊗𝑛h C𝑚 . There is the unitary embedding ⊗𝑛h C𝑚 󴀜󴀠 ⊗𝑛h E. If 𝑚 = 1, then ⊗𝑛h C = C.
For any finite sum 𝜓𝑛 = ∑𝑗 𝑦1𝑗 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑦𝑛𝑗 from the space
⊗𝑛h C𝑚 (or ⊗𝑛h E), we can to define the finite type 𝑛-homogeneous Hilbert-Schmidt polynomials:
𝑛
C𝑚 ∋ 𝑥 󳨃󳨀→ ⟨𝑥⊗𝑛 | 𝜓𝑛 ⟩⊗𝑛 C𝑚 = ∑∏⟨𝑥 | 𝑦𝑡𝑗 ⟩C𝑚 .
h
𝑗 𝑡=1
(33)
in C𝑚 ,
E (E) = ⋃ {E (C𝑚 ) : 𝑚 ∈ N}
in E,
(34)
⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞𝑙
0, . . . , 0, 1 , 0, . . . , 0)𝑚 .
where e𝑚𝑙 = (⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
If s : {1, . . . , 𝑛} 󳨃→ {s(1), . . . , s(𝑛)} runs over all 𝑛elements permutations S(𝑛), then the symmetric 𝑛th tensor
power ⊙𝑛h C𝑚 is defined to be a codomain of the symmetrization mapping:
⊗𝑛h C𝑚 ∋ 𝑥1 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑥𝑛 󳨃󳨀→ 𝑥1 ⊙ ⋅ ⋅ ⋅ ⊙ 𝑥𝑛 ,
1
𝑥1 ⊙ ⋅ ⋅ ⋅ ⊙ 𝑥𝑛 :=
∑ 𝑥 ⊗ ⋅ ⋅ ⋅ ⊗ 𝑥s(𝑛) ,
𝑛! s∈S(𝑛) s(1)
(35)
which is an orthogonal projector. Similarly, the symmetric
𝑛th tensor power ⊙𝑛h E can be defined. Clearly, ⊙𝑛h C𝑚 is a closed
subspace in ⊙𝑛h E.
Given a pair of numbers (𝑚, 𝑛) ∈ N × Z+ , we consider
the 𝑛-fold tensor power of the canonical mapping 𝜋𝑚 : U ∋
𝑢 󳨃→ 𝜋𝑚 (𝑢) ∈ 𝑈(𝑚),
U ∋ 𝑢 󳨃󳨀→
⊗𝑛
𝜋𝑚
(𝑢) ∈
L (⊙𝑛h C𝑚 ) ,
(36)
⊗𝑛
where 𝜋𝑚
(𝑢) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝜋𝑚 (𝑢) ⊗ ⋅ ⋅ ⋅ ⊗ 𝜋𝑚 (𝑢) . If 𝑛 = 0, we put
𝑛
⊗0
𝜋𝑚 (𝑢) = 1 for all 𝑢 ∈ U and 𝑚 ∈ N. The mapping (36) is
Borel and has a continuous restriction to U󸀠 , because 𝜋𝑚 has
the same property (see Section 2).
Let a𝑚 ∈ C𝑚 be an arbitrary fixed element such that
𝑛 𝑚
‖a𝑚 ‖C𝑚 = 1. Then, a⊗𝑛
𝑚 ∈ ⊙h C . Using the mapping (36),
we can write
⊗𝑛
[𝜋𝑚
(𝑢)] (a⊗𝑛
𝑚)
= ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
[𝜋𝑚 (𝑢)] (a𝑚 ) ⊗ ⋅ ⋅ ⋅ ⊗ [𝜋𝑚 (𝑢)] (a𝑚 ).
𝑛
(37)
To any 𝑛-homogeneous Hilbert-Schmidt polynomial (33),
there corresponds the function
𝜓𝑛∗
(𝑢) :=
⊗𝑛
⟨[𝜋𝑚
𝑛
(𝑢)] (a⊗𝑛
𝑚)
| 𝜓𝑛 ⟩⊗𝑛 C𝑚
h
= ∑∏⟨[𝜋𝑚 (𝑢)] (a𝑚 ) | 𝑦𝑡𝑗 ⟩C𝑚
𝑗 𝑡=1
Definition 3. We define P𝑛h (C𝑚 ) to be the space of all functions 𝜓𝑛∗ of the variable 𝑢 ∈ U, determined by the finite type
𝑛-homogeneous Hilbert-Schmidt polynomials (33).
Lemma 4. For any element a𝑚 ∈ C𝑚 such that ‖a𝑚 ‖C𝑚 = 1
the set
S𝑚 = {𝑥 = [𝜋𝑚 (𝑢)] (a𝑚 ) : 𝑢 ∈ U}
(39)
coincides with the unit sphere in C𝑚 . As a consequence, the
one-to-one antilinear corresponding
Consider the canonical orthonormal bases:
E (C𝑚 ) = {e𝑚1 , . . . , e𝑚𝑚 }
of the variable 𝑢 ∈ U. Any cylindrical function of the form
U ∋ 𝑢 󳨃→ ⟨[𝜋𝑚 (𝑢)](a𝑚 ) | 𝑦𝑡𝑗 ⟩C𝑚 has a continuous bounded
restriction to U󸀠 . Therefore, it is 𝜒-essentially bounded on U,
because U \ U󸀠 is a 𝜒-negligible set. Consequently, 𝜓𝑛∗ ∈ 𝐿∞
𝜒
and 𝜓𝑛∗ |U󸀠 is continuous and bounded.
(38)
⊙𝑛h C𝑚 ∋ 𝜓𝑛 󴀘󴀯 𝜓𝑛∗ ∈ P𝑛h (C𝑚 ) .
(40)
Holds, and any function 𝜓𝑛∗ is independent of the choice of an
element a𝑚 ∈ S𝑚 .
Proof. Suppose, on the contrary, that there is an element 𝜓𝑛 ∈
⊙𝑛h C𝑚 such that ⟨𝑥⊗𝑛 | 𝜓𝑛 ⟩⊗𝑛h C𝑚 = 0 for all 𝑥 = [𝜋𝑚 (𝑢)](a𝑚 ) ∈
S𝑚 with 𝑢 ∈ U. The mapping
𝜋𝑚 : U ∋ 𝑢 󳨃󳨀→ 𝜋𝑚 (𝑢) ∈ 𝑈 (𝑚)
(41)
is surjective by surjectivity of the mapping 𝜋𝑚 (see [2,
Lemma 3.1]). Hence, the set S𝑚 coincides with the unit sphere
in C𝑚 and is independent on the choice of an element a𝑚 . By
𝑛-homogeneity, we have ⟨𝑥⊗𝑛 | 𝜓𝑛 ⟩⊗𝑛h C𝑚 = 0 for all 𝑥 ∈ C𝑚 .
Apply the following polarization formula for symmetric
tensor products (see, e.g., [10, Section 1.5]):
𝑧1 ⊙ ⋅ ⋅ ⋅ ⊙ 𝑧𝑛 =
1
∑ ∑ 𝛿 ⋅ ⋅ ⋅ 𝛿𝑛 𝑥⊗𝑛 ,
2𝑛 𝑛! 1≤𝑡≤𝑛 𝛿 =±1 1
(42)
𝑡
with 𝑥 = ∑𝑛𝑡=1 𝛿𝑡 𝑧𝑡 ∈ C𝑚 , which is valid for all 𝑧1 , . . . , 𝑧𝑛 ∈
C𝑚 . It follows that ⟨𝑧1 ⊙ ⋅ ⋅ ⋅ ⊙ 𝑧𝑛 | 𝜓𝑛 ⟩⊗𝑛h C𝑚 = 0 for all
elements 𝑧1 , . . . , 𝑧𝑛 ∈ C𝑚 . Hence, 𝜓𝑛 = 0, because the subset
of all elements 𝑧1 ⊙⋅ ⋅ ⋅⊙𝑧𝑛 is total in ⊙𝑛h C𝑚 . As a consequence,
the subset
⊗𝑛
{𝑥⊗𝑛 = [𝜋𝑚
(𝑢)] (a⊗𝑛
𝑚 ) : 𝑢 ∈ U}
(43)
is also total in ⊙𝑛h C𝑚 . It immediately yields the correspondence (40).
Consider the symmetric Fock space F and its closed
subspace F𝑚 , where
F := C ⊕ E ⊕ (⊙2h E) ⊕ (⊙3h E) ⊕ ⋅ ⋅ ⋅ ,
F𝑚 := C ⊕ C𝑚 ⊕ (⊙2h C𝑚 ) ⊕ (⊙3h C𝑚 ) ⊕ ⋅ ⋅ ⋅ .
(44)
Abstract and Applied Analysis
5
of the variable 𝑢 ∈ U, where we take a𝑚 = e𝑚1 . Consider the
system of functions of the variable 𝑢 ∈ U,
We will use the following notations:
(𝑚) := (𝑚1, . . . , 𝑚𝑚) ,
∗𝑘
𝑚∈N
(45)
󵄨󵄨
󵄨
󵄨󵄨𝑘(𝑚) 󵄨󵄨󵄨 := 𝑘𝑚1 + ⋅ ⋅ ⋅ + 𝑘𝑚𝑚 ,
∗𝑘
As is well known (see, e.g., [11]), the system of symmetric
tensor elements, indexed by the set 𝑘(𝑚) ,
⊗𝑘
𝑚𝑚
E (⊙𝑛h C𝑚 ) = {e(𝑚)(𝑚) = e𝑚1𝑚1 ⊙ ⋅ ⋅ ⋅ ⊙ e⊗𝑘
𝑚𝑚 :
(46)
󵄨󵄨
󵄨󵄨
𝑘(𝑚) ∈ Z𝑚
+ ; 󵄨󵄨𝑘(𝑚) 󵄨󵄨 = 𝑛}
⊙𝑛h C𝑚 ⊂ F𝑚 .
(47)
We will also use the following notations:
[𝑚] := {(11) , (21, 22) , . . . , (𝑚1, . . . , 𝑚𝑚)} ,
𝑚
{𝑘} := {𝑘(1) , . . . , 𝑘(𝑚) } ∈⨉
𝑟=1
Z𝑟+ ,
(48)
Then, the system of symmetric tensor elements with a fixed 𝑛,
indexed by the sets [𝑚] and {𝑘},
⊗𝑘
(1)
(𝑚)
E𝑛 = ⋃ {e⊗{𝑘}
[𝑚] = e(1) ⊙ ⋅ ⋅ ⋅ ⊙ e(𝑚) :
𝑚∈N
⊗𝑘
|𝑘
|
e(1)(1) ∈ E (⊙h (1) C) , . . . , e(𝑚)(𝑚) ∈ E (⊙h (𝑚) C𝑚 )
with fixed |{𝑘}| = 𝑛} ,
forms an orthogonal basis in the subspace ⊙𝑛h E ⊂ F. Thus, the
system
E = {E𝑛 : 𝑛 ∈ Z+ }
E∗𝑚,𝑛 ⊂ P𝑛h (C𝑚 ) ,
(51)
⊗𝑘
⊗𝑛
(𝑚)
e(𝑚)(𝑚) (𝑢) := ⟨[𝜋𝑚
(𝑢)] (e⊗𝑛
𝑚1 ) | e(𝑚) ⟩
⊗𝑛h C𝑚
𝑟=1
(54)
𝑘
e𝑚𝑟 ⟩C𝑚𝑟𝑚 ,
𝑓 ∈ L∞
𝜒 .
(55)
Denote by H2𝜒𝑚 the 𝐿2𝜒 -closure of complex linear spans
of the subsystem E∗𝑚 . As is well known (see, e.g., [12, Theorem
5.6.8]), the space H2𝜒𝑚 is isomorphic to the classic Hardy
space H2𝜒𝑚 (B𝑚 ) of analytic complex functions on the open
unit ball B𝑚 = {𝑥𝑚 ∈ C𝑚 : ‖𝑥𝑚 ‖C𝑚 < 1}. Therefore, the following more general definition seems natural (see, also [8]).
Definition 5. The Hardy-type space H2𝜒 on the space of
virtual unitary matrices U is defined to be the 𝐿2𝜒 -closure of
the complex linear span of the system E∗ .
Theorem 6. The system E∗ of all functions e∗{𝑘}
[𝑚]
∗𝑘
e(𝑚)(𝑚)
∗𝑘
e(𝑟)(𝑟)
=
E∗𝑟,|𝑘(𝑟) |
⋅ ⋅⋅⋅ ⋅
with 𝑚 ∈ N, such that
∈
as
𝑟 = 1, . . . , 𝑚, forms an orthogonal basis in the Hardy-type
spaces H2𝜒 with norms
𝑚
(𝑟 − 1)!(𝑘)𝑟 !
󵄩󵄩 ∗{𝑘} 󵄩󵄩
󵄩󵄩e[𝑚] 󵄩󵄩 2 = (∏
󵄨󵄨 )
󵄨󵄨
󵄩𝐿 𝜒
󵄩
𝑟=1 (𝑟 − 1 + 󵄨󵄨(𝑘)𝑟 󵄨󵄨)!
1/2
.
(56)
Proof. If |{𝑘}| ≠ |{𝑞}|, then from (28), it follows that
∗{𝑞}
∫ e∗{𝑘}
[𝑚] ⋅ e[𝑛] 𝑑𝜒
U
∗{𝑞}
= ∫ e∗{𝑘}
[𝑚] (exp (i𝜗) 𝑢) ⋅ e[𝑛] (exp (i𝜗) 𝑢) 𝑑𝜒 (𝑢)
U
𝜋
1
∗{𝑞}
󵄨󵄨 󵄨󵄨
∫ e∗{𝑘}
[𝑚] e[𝑛] 𝑑𝜒 ∫ exp (i (|{𝑘}| − 󵄨󵄨{𝑞}󵄨󵄨) 𝜗) 𝑑𝜗
2𝜋 U
−𝜋
= 0.
(57)
=
of the following 𝜒𝑚 -integrable cylindrical functions:
= ∏⟨(𝜋𝑚 ∘ 𝑢) (e𝑚1 ) |
E∗𝑚 = {E∗𝑚,𝑛 : 𝑛 ∈ Z+ } .
Let 𝐿2𝜒 be the space of square 𝜒-integrable complex functions , 𝑓 on the space of virtual matrices U. Since 𝜒 is a
2
probability measure, the embedding L∞
𝜒 ⊂ 𝐿 𝜒 holds and
(50)
forms an orthogonal basis in the symmetric Fock space F.
By virtue of the one-to-one mapping (40), the system
of symmetric tensor elements E(⊙𝑛h C𝑚 ) uniquely defines the
following corresponding system:
𝑚
E∗ = {E∗𝑛 : 𝑛 ∈ Z+ } ,
∗𝑘
e(1)(1)
(49)
∗𝑘
generated by the system of symmetric tensor elements E𝑛 . All
these functions belong to the space L∞
𝜒 by their definition.
Denote
𝑢∈U
{𝑘}! := 𝑘(1) ! ⋅ . . . ⋅ 𝑘(𝑚) !.
|𝑘 |
with fixed |{𝑘}| = 𝑛} ,
󵄨
󵄩󵄩 󵄩󵄩
󵄨
󵄩󵄩𝑓󵄩󵄩𝐿2𝜒 ≤ ess sup 󵄨󵄨󵄨𝑓 (𝑢)󵄨󵄨󵄨 ,
󵄨
󵄨 󵄨
󵄨
|{𝑘}| := 󵄨󵄨󵄨𝑘(1) 󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + 󵄨󵄨󵄨𝑘(𝑚) 󵄨󵄨󵄨 ,
⊗𝑘
(53)
6. The Hardy-Type Space
forms an orthogonal basis in the subspace
⊗𝑘
∗𝑘
e(1)(1) ∈ E∗1,|𝑘(1) | , . . . , e(𝑚)(𝑚) ∈ E∗𝑚,|𝑘(𝑚) |
𝑘(𝑚) ! := 𝑘𝑚1 ! ⋅ . . . ⋅ 𝑘𝑚𝑚 !.
⊗𝑘
∗𝑘
(1)
(𝑚)
E∗𝑛 = ⋃ {e∗{𝑘}
[𝑚] = e(1) ⋅ ⋅ ⋅ ⋅ ⋅ e(𝑚) :
𝑘(𝑚) := (𝑘𝑚1 , . . . , 𝑘𝑚𝑚 ) ∈ Z𝑚
+,
(52)
∗{𝑞}
2
So, e∗{𝑘}
[𝑚] ⊥ e[𝑛] in the space 𝐿 𝜒 if |{𝑘}| ≠ |{𝑞}| for all indices
[𝑚], [𝑛].
6
Abstract and Applied Analysis
Let |{𝑘}| = |{𝑞}| and 𝑚 > 𝑛 for definiteness. If the
∗{𝑞}
elements e∗{𝑘}
[𝑚] and e[𝑛] are different, then there exists a
subindex 𝑚𝑠 ∈ {11, 21, 22, . . ., 𝑚1, . . ., 𝑚𝑚} in the blockindex [𝑚] = [(11), (21, 22), . . ., (𝑚1, . . ., 𝑚𝑚)] such that 𝑚𝑠 ∉
{11, 21, 22, . . . , 𝑛1, . . . , 𝑛𝑛}, where [𝑛] = [(11), (21, 22), . . .,
(𝑛1, . . . , 𝑛𝑛)]. The formula (28) implies that for the group of
shifts 𝑄𝑔𝑚𝑠 (𝜗) generated by elements 𝑔𝑚𝑠 (𝜗) ∈ 𝑈𝑠2 (𝑚) with the
subindex 𝑚𝑠,
∗{𝑞}
∫ e∗{𝑘}
[𝑚] ⋅ e[𝑛] 𝑑𝜒
U
=∫
U
=
󵄨
󵄨󵄨2
󵄨
∫ 󵄨󵄨󵄨󵄨e∗{𝑘}
[𝑚] 󵄨󵄨 𝑑𝜒
U
𝑚
= ∫ 𝑑𝜒 (𝑢) ∏ ∫
U
𝑟=1
𝑈2
(𝑟)
󵄨 ∗(𝑘) 󵄨2
𝑄𝑔𝑟 󵄨󵄨󵄨󵄨e(𝑟) 𝑟 󵄨󵄨󵄨󵄨 (𝑢) 𝑑 (𝜒𝑟 ⊗ 𝜒𝑟 ) (𝑔𝑟 )
𝑚 󵄩
󵄩 ∗𝑘 󵄩󵄩2
= ∏󵄩󵄩󵄩e(𝑟)(𝑟) 󵄩󵄩󵄩 2 ,
󵄩
󵄩𝐿 𝜒𝑟
𝑟=1
(62)
𝑄𝑔𝑚𝑠 (𝜗) e∗{𝑘}
[𝑚]
⋅
∗{𝑞}
𝑄𝑔𝑚𝑠 (𝜗) e[𝑛] 𝑑𝜒
(58)
𝜋
1
∗{𝑞}
∫ e∗{𝑘} ⋅ e 𝑑𝜒 ∫ exp (i𝑘𝑚𝑠 𝜗) 𝑑𝜗 = 0.
2𝜋 U [𝑚] [𝑛]
−𝜋
because ∫U 𝑑𝜒 = 1. It follows that
𝑚 󵄩
𝑚
󵄩2
󵄩󵄩 ∗{𝑘} 󵄩󵄩2
𝑟!
(𝑟) 󵄩
󵄩󵄩 = ∏ (𝑟 − 1)!(𝑘)
󵄩󵄩e[𝑚] 󵄩󵄩 2 = ∏󵄩󵄩󵄩󵄩e∗𝑘
,
󵄨
󵄩
(𝑟)
2
󵄩
󵄩𝐿 𝜒
󵄨
󵄩
󵄩𝐿 𝜒𝑟 𝑟=1 (𝑟 − 1 + 󵄨󵄨(𝑘)𝑟 󵄨󵄨󵄨󵄨) !
𝑟=1
∗{𝑞}
2
Hence, e∗{𝑘}
[𝑚] ⊥ e[𝑛] in 𝐿 𝜒 .
∗𝑘
e∗{𝑘}
[𝑚]
∗{𝑞}
≠ e[𝑛] ,
then
Let now |{𝑘}| = |{𝑞}| and 𝑚 = 𝑛. If
{𝑘} ≠ {𝑞}. Hence, there exists a sub-index 𝑟𝑠 in the blockindex [𝑚] = [𝑛] such that 𝑘𝑟𝑠 ≠ 𝑞𝑟𝑠 . Similarly as previous
mentioned, applying the formula (28) to the group of shifts
𝑄𝑔𝑟𝑠 (𝜗) generated by elements 𝑔𝑟𝑠 (𝜗) ∈ 𝑈𝑠2 (𝑟) with the subindex 𝑟𝑠, we get
∗{𝑞}
∫ e∗{𝑘}
[𝑚] ⋅ e[𝑛] 𝑑𝜒
U
=
by the well-known formula [12, Section 1.4.9]. Using the
formula (27) 𝑚-times for 𝑟 = 1, . . . , 𝑚, we get
𝜋
1
∗{𝑞}
∫ e∗{𝑘}
[𝑚] e[𝑛] 𝑑𝜒 ∫ exp (i (𝑘𝑟𝑠 − 𝑞𝑟𝑠 ) 𝜗) 𝑑𝜗
2𝜋 U
−𝜋
= 0.
(59)
∗𝑘
(𝑚)
(𝑚)
for all e∗{𝑘}
[𝑚] = e(1) ⋅ ⋅ ⋅ ⋅ ⋅ e(𝑚) .
As is known (see, e.g., [11]), the system E𝑚 of symmetric
⊗(𝑘)
tensors e(𝑚) 𝑚 with a fixed 𝑚 forms an orthogonal basis
⊗(𝑘)
in the symmetric Fock space F𝑚 with norms ‖e(𝑚) 𝑚 ‖F
√(𝑘)𝑚 !/|(𝑘)𝑚 |!. Similarly, the system E of symmetric tensors
⊗𝑘
⊗(𝑘)
⊗(𝑘)1
e⊗{𝑘}
⊙ ⋅ ⋅ ⋅ ⊙ e(𝑚)(𝑚) with all 𝑚 ∈ N, such that e(𝑟) 𝑟 ∈
[𝑚] = e(1)
E𝑟,|(𝑘)𝑟 | as 𝑟 = 1, . . . , 𝑚, forms an orthogonal basis in the
√
symmetric Fock space F with norms ‖e⊗{𝑘}
[𝑚] ‖F = {𝑘}!/|{𝑘}|!.
Combining Lemma 4, Theorem 6, and [12, Theorem
5.6.8], we obtain the following.
Theorem 7. Antilinear extensions of the one-to-one mappings
between the orthonormal bases
𝑈2 (𝑟)
∗(𝑘)
e(𝑚) 𝑚
e(𝑚) 𝑚
󵄩󵄩 ⊗(𝑘)𝑚 󵄩󵄩 󴀘󴀯 󵄩󵄩 ∗(𝑘)𝑚 󵄩󵄩 ,
󵄩󵄩e(𝑚) 󵄩󵄩
󵄩󵄩e(𝑚) 󵄩󵄩 2
󵄩
󵄩F𝑚
󵄩
󵄩𝐿 𝜒𝑚
∗{𝑞}
e[𝑛]
Hence, in this case also
⊥
under the measure 𝜒.
Let 𝑔𝑟 = (1𝑟 , 𝑤𝑟 ) ∈ 𝑈 (𝑟) and 𝑢 ∈ U. Using (11) and (52),
we have
∫
=
𝑚
⊗(𝑘)
e∗{𝑘}
[𝑚]
2
(63)
(64)
e⊗{𝑘}
e∗{𝑘}
[𝑚]
[𝑚]
󵄩󵄩 ⊗{𝑘} 󵄩󵄩 󴀘󴀯 󵄩󵄩 ∗{𝑘} 󵄩󵄩 ,
󵄩󵄩e[𝑚] 󵄩󵄩 2
󵄩󵄩󵄩e[𝑚] 󵄩󵄩󵄩F
󵄩
󵄩𝐿 𝜒
󵄨 ∗(𝑘) 󵄨2
𝑄𝑔𝑟 󵄨󵄨󵄨󵄨e(𝑟) 𝑟 󵄨󵄨󵄨󵄨 (𝑢) 𝑑 (𝜒𝑟 ⊗ 𝜒𝑟 ) (𝑔𝑟 )
𝑟 󵄨
𝑘𝑟𝑙 󵄨󵄨2
󵄨
∏󵄨󵄨󵄨󵄨⟨[𝑤𝑟−1 𝜋𝑟 (𝑢)] (e𝑟1 ) | e𝑟𝑙 ⟩C𝑟 󵄨󵄨󵄨󵄨 𝑑𝜒𝑟 (𝑤𝑟 ) .
󵄨
𝑈(𝑟) 𝑙=1 󵄨
(60)
uniquely define the corresponding anti-linear isometric isomorphisms
However, the previous integral with the Haar measure 𝜒𝑟 is
independent of 𝜋𝑟 (𝑢) ∈ 𝑈(𝑟). It follows that
Reasoning by analogy with [8, Proposition 6.1 and Theorem 7.1], it is easy to show that the Hardy space H2𝜒 possesses
the reproducing kernel of a Cauchy type
=∫
∫
𝑈2 (𝑟)
󵄨 ∗(𝑘) 󵄨2
𝑄𝑔𝑟 󵄨󵄨󵄨󵄨e(𝑟) 𝑟 󵄨󵄨󵄨󵄨 (𝑢) 𝑑 (𝜒𝑟 ⊗ 𝜒𝑟 ) (𝑔𝑟 )
𝑟 󵄨
𝑘𝑟𝑙 󵄨󵄨2
󵄨
∏󵄨󵄨󵄨󵄨⟨𝑤𝑟−1 (e𝑟1 ) | e𝑟𝑙 ⟩C𝑟 󵄨󵄨󵄨󵄨 𝑑𝜒𝑟 (𝑤𝑟 )
󵄨
𝑈(𝑟) 𝑙=1 󵄨
=∫
(𝑟 − 1)!(𝑘)𝑟 !
󵄩󵄩 ∗(𝑘)𝑟 󵄩󵄩2
=
󵄨󵄨 = 󵄩󵄩󵄩e(𝑟) 󵄩󵄩󵄩𝐿2𝜒
󵄨󵄨
(𝑟 − 1 + 󵄨󵄨(𝑘)𝑟 󵄨󵄨) !
𝑟
(61)
F𝑚 ≃ H2𝜒𝑚 (B𝑚 ) ,
F ≃ H2𝜒 .
(65)
∗{𝑘}
e∗{𝑘}
[𝑚] (V) e[𝑚] (𝑢)
C (V, 𝑢) = ∑ ∑
󵄩󵄩 ∗{𝑘} 󵄩󵄩2
󵄩󵄩e[𝑚] 󵄩󵄩 2
𝑛∈Z+ |{𝑘}|=𝑛
󵄩
󵄩𝐿 𝜒
∞
−𝑚
= ∏ (1 − ⟨(𝜋𝑚 ∘ V) (e𝑚1 ) | (𝜋𝑚 ∘ 𝑢) (e𝑚1 )⟩E )
,
𝑚=1
(66)
Abstract and Applied Analysis
7
with 𝑢, V ∈ U, where the sum ∑|{𝑘}|=𝑛 is over all indices {𝑘} ∈
𝑟
{⨉𝑚
𝑟=1 Z+ : 𝑚 ∈ N} such that |{𝑘}| = 𝑛. As a consequence, the
integral representation of any function 𝑓 ∈ H2𝜒 ,
𝑓 (𝜆V) = ∫ 𝑓 (𝑢) C (𝜆V, 𝑢) 𝑑𝜒 (𝑢)
U
(67)
gives a unique analytic extension in the complex variable 𝜆 ∈
B1 for all elements V ∈ U such that
󵄩
󵄩2
∑ 𝑚󵄩󵄩󵄩(𝜋𝑚 ∘ V) (e𝑚1 )󵄩󵄩󵄩C𝑚 < ∞.
𝑚∈N
(68)
References
[1] Y. A. Neretin, “Hua-type integrals over unitary groups and over
projective limits of unitary groups,” Duke Mathematical Journal,
vol. 114, no. 2, pp. 239–266, 2002.
[2] G. Olshanski, “The problem of harmonic analysis on the
infinite-dimensional unitary group,” Journal of Functional Analysis, vol. 205, no. 2, pp. 464–524, 2003.
[3] D. Pickrell, “Measures on infinite dimensional Grassmann
manifolds,” Journal of Functional Analysis, vol. 70, no. 2, pp.
323–356, 1987.
[4] S. Kerov, G. Olshanski, and A. Vershik, “Harmonic analysis
on the infinite symmetric group: a deformation of regular
representation,” Comptes Rendus de l’Académie des Sciences.
Series I, vol. 316, pp. 773–778, 1993.
[5] A. Borodin and G. Olshanski, “Harmonic analysis on the
infinite-dimensional unitary group and determinantal point
processes,” Annals of Mathematics, vol. 161, no. 3, pp. 1319–1422,
2005.
[6] E. Tomas, “On Prohorov’s criterion for projective limits,” Operator Theory, vol. 168, pp. 251–261, 2006.
[7] Y. Yamasaki, “Projective limit of Haar measures on O(n),”
Research Institute for Mathematical Sciences, vol. 8, pp. 141–149,
1972/73.
[8] O. Lopushansky and A. Zagorodnyuk, “Hardy type spaces
associated with compact unitary groups,” Nonlinear Analysis:
Theory, Methods and Applications, vol. 74, no. 2, pp. 556–572,
2011.
[9] N. Bourbaki, Integration II, Springer, Berlin, Germany, 2004.
[10] K. Floret, “Natural norms on symmetric tensor products of
normed spaces,” Note di Matematica, vol. 17, pp. 153–188, 1997.
[11] M. Reed and B. Simon, Methods of Modern Mathematical
Physics, Volume II, Academic Press, New York, NY, USA, 1975.
[12] W. Rudin, Function Theory in the Unit Ball of C𝑛 , Springer,
Berlin, Germany, 1980.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Mathematical Problems
in Engineering
Journal of
Mathematics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Discrete Dynamics in
Nature and Society
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Abstract and
Applied Analysis
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014