Wiener's Lemma for Infinite Matrices II

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by  Qiyu Sun

WIENER’S LEMMA FOR INFINITE MATRICES II
QIYU SUN
0
1
0
2
Abstract. Inthispaper,weintroduceaclassofinfinitematrices
n
related to the Beurling algebraof periodic functions, and we show
a
J that it is an inverse-closed subalgebra of (ℓq), the algebra of all
9 bounded linearoperatorsonthe weightseBquewnce spaceℓqw,forany
1 q < and any discrete Muckenhoupt A -weight w.
] ≤ ∞ q
A
F 1. Introduction
.
h
t Let us begin the sequel to [43] by introducing a new class of infinite
a
matrices,
m
1 [ (1.1) B(Zd,Zd) := nA := (a(i,j))i,j∈Zd (cid:12) kAkB(Zd,Zd) < ∞o,
v where d 1, x := max( x ,..., x ) fo(cid:12)(cid:12)r x := (x ,...,x ) Rd,
7 ≥ | |∞ | 1| | d| 1 d ∈
5 and
4
1 (1.2) A := sup a(i,j) .
(Zd,Zd)
1. k kB kXZd(cid:16)|i−j|∞≥|k|∞| |(cid:17)
0 ∈
0 It is observed that a Toeplitz matrix A := (a(i j))i,j Z associated
v:1 wFoituhriearsseeqruieesnaˆc(eξ)a::== (a(n)a)n(∈nZ)ebxeplo(ng√s to1 Bnξ(Z) ,bZe−l)onifgsant∈odtohnelyalgifebthrae
i n Z − −
X P ∈
∞ ∞
r (1.3) A (T) := a(n)e √ 1 nξ sup a(n) < .
a ∗ − −
nnX= (cid:12)(cid:12) Xk=0|n|≥k| | ∞o
−∞
(cid:12)
The above algebra A (T) was introduced by A. Beurling for establish-
∗
ing contraction properties of periodic functions [8], and was used in
considering pointwise summability of Fourier series [10, 15, 16, 40, 48].
So the class (Zd,Zd) of infinite matrices can be interpreted as a non-
commutativeBmatrix extension of the Beurling algebra A (T).
∗
Date: January 9, 2010.
1991 Mathematics Subject Classification. 46H10, 42B25, 47B35, 47B37, 47A10,
46A45, 41A15, 42C40,94A12 .
Key words and phrases. Wiener’s lemma, Beurling algebra, Muckenhoupt A -
q
weight, infinite matrix, stability, left inverse.
1
2 QIYUSUN
Define the Gr¨ochenig-Schur class (Zd,Zd) by
S
(Zd,Zd) := a(i,j)
S n(cid:0) (cid:1)i,j∈Zd (cid:12)
(cid:12)
(1.4) sup a(i(cid:12),j) ,sup a(i,j) <
| | | | ∞
(cid:16)i∈ZdjXZd j∈ZdiXZd (cid:17) o
∈ ∈
[25, 37, 41, 43], and the Gohberg-Baskakov-Sj¨ostrand class (Zd,Zd) by
C
(1.5) (Zd,Zd) := a(i,j) sup a(i,j) <
C i,j Zd | | ∞
n(cid:0) (cid:1) ∈ (cid:12)(cid:12) kXZd(cid:0)i−j=k (cid:1) o
(cid:12) ∈
[6, 20, 25, 32, 39, 43]. The above two classes of infinite matrices ap-
pearedinthestudy ofGabortime-frequency analysis, nonuniformsam-
pling, and algebra of pseudo-differential operators etc (see [2, 4, 24, 28,
39, 42] for a sample of papers). From (1.1), (1.4) and (1.5) it follows
that
(1.6) (Zd,Zd) (Zd,Zd) (Zd,Zd).
B ⊂ C ⊂ S
This shows that any matrix in (Zd,Zd) belongs to the Gro¨chenig-
Schur class (Zd,Zd) and also thBe Gohberg-Baskakov-Sj¨ostrand class
(Zd,Zd). S
C An equivalent way of defining (Zd,Zd) is the existence of a radially
B
decreasingsequence b(i) foranyinfinitematrixA := (a(i,j))
i Zd i,j Zd
(Zd,Zd) such that{ }∈ ∈ ∈
B
(1.7) a(i,j) b(i j) for all i,j Zd,
| | ≤ − ∈
(1.8) b(i) < ,
∞
iXZd
∈
and
(1.9) b(i) = h( i ) for some decreasing sequence h(n) .
| |∞ { }∞n=0
Therefore any infinite matrix in (Zd,Zd) is dominated by a convo-
B
lution operator associated with a summable, radial and (radially) de-
creasing sequence. We remark that any infinite matrix in the Gohberg-
Baskakov-Sj¨ostrand class (Zd,Zd) is dominated by a convolution op-
C
erator associated with a summable sequence [6, 20, 25, 32, 39, 43].
A positive sequence w := (w(i)) is said to be a discrete A -weight
i Zd q
∈
for 1 < q < if there exists a positive constant C such that
∞
(1.10)
1/(q 1)
N d w(i) N d (w(i)) (q 1) − C
− − − −
≤
(cid:16) i a+X[0,N 1]d (cid:17)(cid:16) i a+X[0,N 1]d (cid:17)
∈ − ∈ −
WIENER’S LEMMA FOR INFINITE MATRICES II 3
hold for all a Zd and 1 N Z, and to be a discrete A -weight if
1
∈ ≤ ∈
there exists a positive constant C such that
(1.11) N d w(i) C inf w(i)
−
X ≤ i a+[0,N 1]d
i a+[0,N 1]d ∈ −
∈ −
holdfor all a Zd and 1 N Z [18, 40]. The smallest constant C for
∈ ≤ ∈
which (1.10) holds when 1 < q < , and respectively for which (1.11)
∞
holds when q = 1, to be denoted by A (w), is the discrete A -bound.
q q
The positive sequences ((1 + i )α) with d < α < d(q 1) if
i Zd
| |∞ ∈ − −
1 < q < , and respectively with d < α 0 if q = 1, are discrete
∞ − ≤
A -weights.
q
For 1 < q < , a positive locally integrable function w on Rd is said
∞
to be an A -weight if there exists a positive constant C such that
q
1 1 1/(q 1)
(1.12) w(x)dx w(x) (q 1)dx − C
− −
(cid:16)|Q| ZQ (cid:17)(cid:16)|Q| ZQ (cid:17) ≤
for all cubic Q Rd, where Q denotes the Lebesgue measure of the
⊂ | |
cubic Q [36]. Similarly for q = 1, a positive locally integrable function
w is said to be an A -weight if there exists a positive constant C such
1
that
1
(1.13) w(y)dy Cw(x), x Q
Q Z ≤ ∈
| | Q
for all cubic Q Rd [14]. One may then verify that a positive se-
⊂
quence w := w(i) is a discrete A -weight if and only if w˜(x) :=
i Zd q
w(i)χ (cid:0) (cid:1)∈(x i) is an A -weight, where 1 q < and χ
Pis ti∈hZedcharac[t−e1r/i2s,1t/ic2)dfunc−tion on a seqt E [33]. ≤ ∞ E
For 1 q < and a positive sequence w := (w(i)) on Zd, let
i Zd
ℓq := ℓq (≤Zd) be∞the space of all weighted q-summable seq∈uences on Zd,
w w
i.e.,
1/q
(1.14) ℓq (Zd) := (c(i)) c := c(i) qw(i) < .
w n i∈Zd (cid:12)(cid:12) k kq,w (cid:16)iXZd| | (cid:17) ∞o
(cid:12) ∈
For the trivial weight w (i.e. w (i) = 1 for all i Zd), we will use ℓq
0 0
∈
and instead of ℓq and for brevity. Define the discrete
k · kq w0 k · kq,w0
maximal function by
(1.15)
1
Mc(i) := sup c(k) for c := (c(i)) .
0 N Z (2N +1)d X | | i∈Zd
≤ ∈ k i+[ N,N]d
∈ −
Similar to the characterization of an A -weight on Rd via the standard
q
maximal operator [34], the discrete A -weight can be characterized via
q
4 QIYUSUN
thediscrete maximal functionontheweighted ℓq space. Moreprecisely,
a positive sequence w := (w(i)) is a discrete A -weight if and only
i Zd q
if the discrete maximal operator∈c Mc is of weak-type (ℓq,ℓq), i.e.,
7−→ w w
there exists a positive constant C such that
C
(1.16) w(i) c q for all α > 0 and c ℓp.
≤ αqk kq,w ∈ w
X
Mc(i) α
≥
Moreover for 1 < q < , the discrete maximal operator M is of strong
∞
type (ℓq,ℓq ) for a discrete A -weight w, i.e., there exists a positive
w w q
constant C such that
′
(1.17) Mc C c for all c ℓp.
k kq,w ≤ ′k kq,w ∈ w
The reader may refer to [18] for a complete account of the theory of
weighted inequalities and its ramification.
Now let us present our results for infinite matrices in (Zd,Zd). In
B
Section 3, we establish the following algebraic properties for the class
(Zd,Zd) of infinite matrices.
B
Theorem 1.1. Let 1 q < , and let w be a discrete A -weight. Then
q
(Zd,Zd) is a unital ≤Banach∞algebra under matrix multiplication, and
B
also a subalgebra of (ℓq ), the algebra of all bounded linear operators
B w
on the weight sequence space ℓq .
w
By Theorem 1.1, every infinite matrix A (Zd,Zd) defines a
∈ B
bounded operator on ℓq for any 1 q < and for any discrete
w ≤ ∞
A -weight w, i.e., there exists a positive constant C such that
q
(1.18) Ac C c for all c ℓq.
k kq,w ≤ k kq,w ∈ w
Besides the above boundedness of an infinite matrix on the weighted
sequence space ℓq , it is natural to consider ℓq-stability. Here for 1
q < and a poswitive sequence w on Zd, we sawy that an infinite matri≤x
∞
A has ℓq -stability if there exists a positive constant C such that
w
(1.19) C 1 c Ac C c for all c ℓq .
− k kq,w ≤ k kq,w ≤ k kq,w ∈ w
The ℓq-stability is one of the basic assumptions for infinite matrices
w
arising in the study of spline approximation, Gabor time-frequency
analysis, nonuniform sampling, and algebra of pseudo-differential op-
erators etc (see [2, 4, 24, 28, 29, 39, 42, 43] and the references therein.)
InSection4, we establish theequivalence ofℓq-stabilities ofanyinfinite
w
matrix in (Zd,Zd) for different exponents 1 q < and for different
B ≤ ∞
discrete A -weights w.
q
WIENER’S LEMMA FOR INFINITE MATRICES II 5
Theorem 1.2. Let A (Zd,Zd). If A has ℓq-stability for some
1 q < and for some∈diBscrete A -weight w, thewn it has ℓq′ -stability
≤ ∞ q w′
for all 1 q′ < and for all discrete Aq′-weights w′.
≤ ∞
The reader may refer to [1, 38, 49] for the equivalence of unweighted
ℓq-stabilityofinfinitematricesintheGohberg-Baskakov-Sj¨ostrandclass
w
(Zd,Zd). If A (ℓq ) hasa left inverseB (ℓq ), i.e., BA = I, then
C ∈ B w ∈ B w
A has ℓq -stability. The converse is not true in general, unless q = 2.
w
As an application of Theorem 1.2, we show that the converse holds for
any infinite matrix A in (Zd,Zd).
B
Corollary 1.3. Let 1 q < , and let w be a discrete A -weight.
q
Then an infinite matrix≤in (Z∞d,Zd) has ℓq -stability if and only if it
B w
has a left inverse in (ℓq ).
B w
Given a Banach algebra , a subalgebra of is said to be inverse-
B A B
closedif A andthe inverse A 1 of the element A exists in implies
−
∈ A B
that A 1 [19, 35, 47]. The next question following the ℓq-stability
of an i−nfi∈nitAe matrix in (Zd,Zd) is whether its inverse, ifwexists in
(ℓq ), belongs to (Zd,ZBd), or in the other word, whether (Zd,Zd)
B w B B
is an inverse-closed subalgebra of (ℓq ).
B w
The inverse-closedness for the subalgebra of absolutely convergent
Fourier series in the algebra of bounded periodic functions was first
studied in [9, 19, 35, 50]. The inverse-closed property (=Wiener’s
lemma) has been established for infinite matrices satisfying various
off-diagonal decay conditions, see [3, 5, 6, 17, 23, 25, 29, 39, 41, 43]
for a sample of papers. Inverse-closedness occurs under various names
(such as spectral invariance, Wiener pair, local subalgebra) in many
fields of mathematics, see the survey [21].
The inverse-closed property for non-commutative matrix subalgebra
has been shown to be crucial for the well-localization of dual wavelet
frames and dual Gabor frames [4, 24, 29, 30, 31], the algebra of pseudo-
differential operators [22, 28, 39], the fast implementation in numerical
analysis [12, 13, 27], and the local reconstruction in sampling theory
[2, 42, 45].
It mixes art and hard mathematical work to consider the inverse-
closed subalgebra of (ℓq ). The art is to guess the off-diagonal decay
B w
of infinite matrices in an algebra , while the work is to prove the
A
inverse-closedness of the algebra in (ℓq ). There are several ap-
A B w
proaches to prove the inverse-closedness of a subalgebra of (ℓq ). Here
B w
arethree of them: (i) the indirect approach, such as the Gelfand’s tech-
nique [17, 19, 25]; (ii) the semi-direct approach, such as the bootstrap
6 QIYUSUN
argument [29] and the derivation trick [23]; (iii) the direct approach,
such as the commutator trick [6, 39] and the asymptotic estimate tech-
nique [38, 41, 43]. Each approach has its advantages and limitations.
For instance, the Gelfand technique and the asymptotic estimate tech-
nique work well for inverse-closed subalgebras of (ℓq) with q = 2, but
B
they are not directly applicable for inverse-closed subalgebras of (ℓq)
B
with q = 2. The commutator trick is applicable to establish Wiener’s
6
lemma for subalgebra of (ℓq),1 q [6, 39]. In Section 5, we
B ≤ ≤ ∞
combine the commutator trick, the asymptotic estimate technique and
the equivalence of ℓq-stability for different exponents q and for dif-
w
ferent discrete A -weights w, and then establish Wiener’s lemma for
q
subalgebras of (ℓq ), where 1 q < and w is a discrete A -weight.
B w ≤ ∞ q
Theorem 1.4. Let 1 q < and let w be a discrete A -weight. Then
q
(Zd,Zd) is an invers≤e-close∞d subalgebra of (ℓq ).
B B w
As an application of Theorem 1.4, we obain Wiener’s lemma for the
Beurling algebra A (T) of periodic functions [7].
∗
Corollary 1.5. If f A (T) and f(ξ) = 0 for all ξ R then 1/f
∗
A (T). ∈ 6 ∈ ∈
∗
As applications of Theorems 1.2 and 1.4, we establish the equiva-
lence between the ℓq -stability of an infinite matrix in (Zd,Zd) and
the existence of its lweft inverse in (Zd,Zd). B
B
Corollary 1.6. Let 1 q , and let w be a discrete A -weight.
q
Then an infinite matrix≤in ≤(Z∞d,Zd) has ℓq -stability if and only if it
has a left inverse in (Zd,ZBd). w
B
2. A class of infinite matrices
In this section, we introduce a class of infinite matrices with off-
diagonal decay, which includes the class (Zd,Zd) in the Introduction
B
as a special case.
A weight matrix on Zd Zd, or a weight matrix for brevity, is a
×
positive matrix u := (u(i,j)) with each entry not less than one,
i,j Zd
∈
i.e.,
(2.1) u(i,j) 1 for all i,j Zd.
≥ ∈
For 1 p and a weight matrix u := (u(i,j)) , define
i,j Zd
≤ ≤ ∞ ∈
(2.2) (Zd,Zd) := A := (a(i,j)) A < ,
Bp,u n i,j∈Zd (cid:12) k kBp,u(Zd,Zd) ∞o
(cid:12)
(cid:12)
WIENER’S LEMMA FOR INFINITE MATRICES II 7
where
(2.3) A := sup a(i,j) u(i,j) .
k kBp,u(Zd,Zd) (cid:13)(cid:13)(cid:16)|i−j|∞≥|k|∞| | (cid:17)k∈Zd(cid:13)(cid:13)p
(cid:13) (cid:13)
If p = 1 and u 1 (i.e., all entries of the weight matrix u are equal
≡
to 1), then
(2.4) (Zd,Zd) = (Zd,Zd) and = .
Bp,u B k·kBp,u(Zd,Zd) k·kB(Zd,Zd)
Inthispaper,weuse , , , insteadof (Zd,Zd), (Zd,Zd),
Bp,u B k·kBp,u k·kB Bp,u B
, for brevity.
k·kBp,u(Zd,Zd) k·kB(Zd,Zd)
Remark 2.1. Let 1 p and u be a weighted matrix. Define the
Gr¨ochenig-Schur clas≤s ≤(∞Zd,Zd) of infinite matrices by
p,u
S
(2.5) (Zd,Zd) := A := (a(i,j)) A < ,
Sp,u n i,j∈Zd (cid:12) k kSp,u(Zd,Zd) ∞o
(cid:12)
where
(cid:12)
(2.6)
A := max sup a(i,j)u(i,j) ,sup a(i,j)u(i,j)
k kSp,u(Zd,Zd) (cid:16)i Zd(cid:13)(cid:0) (cid:1)j∈Zd(cid:13)p j Zd(cid:13)(cid:0) (cid:1)i∈Zd(cid:13)p(cid:17)
∈ (cid:13) (cid:13) ∈ (cid:13) (cid:13)
[25, 37, 41, 43]. For p = 1, the class (Zd,Zd) were introduced by
1,u
S
Schur [37] for weight matrices u := (w(i)/w(j)) generated by pos-
i,j Zd
∈
itive sequences w := (w(i)) , and by Gr¨ochenig and Leinert [25] for
i Zd
∈
weight matrices u := (v(i j)) associated with positive functions
i,j Zd
v on Rd. For 1 p , −the clas∈s (Zd,Zd) was introduced by Sun
p,u
≤ ≤ ∞ S
for polynomial weights u := ((1+ i j )α) with α > d(1 1/p)
i,j Zd
| − |∞ ∈ −
in [41] and for any weighted matrix u in [43]. From the above defini-
tion of the Gr¨ochenig-Schur class (Zd,Zd), the following inclusion
p,u
S
follows:
(2.7) (Zd,Zd) (Zd,Zd)
p,u p,u
B ⊂ S
for any 1 p and for any weight matrix u.
≤ ≤ ∞
Remark 2.2. Let 1 p and u be a weighted matrix. Define the
Gohberg-Baskakov-Sj≤¨ostra≤nd∞class (Zd,Zd) of infinite matrices by
p,u
C
(2.8) (Zd,Zd) := A := (a(i,j)) A < ,
Cp,u n i,j∈Zd (cid:12) k kCp,u(Zd,Zd) ∞o
(cid:12)
where
(cid:12)
(2.9) A := sup a(i,j) u(i,j)
k kCp,u(Zd,Zd) (cid:13)(cid:13)(cid:16)i−j=k(cid:0)| | (cid:1)(cid:17)k∈Zd(cid:13)(cid:13)p
[6, 20, 25, 26, 32, 39, 43]. F(cid:13)or p = 1 and the trivial weig(cid:13)ht matrix u
0
(i.e., u (i,j) = 1for all i,j Zd), theclass (Zd,Zd) was introduced
0 ∈ C1,u0
by Gohberg, Kaashoek, and Woerdeman [20] as a generalization of
8 QIYUSUN
the class of Toeplitz matrices associated with summable sequences. It
was reintroduced by Sj¨ostrand [39] in considering algebra of pseudo-
differential operators. For p = 1 and nontrivial weight matrices u :=
(v(i j)) associated with positive functions v on Rd, the class
i,j Zd
(−Zd,Zd)∈was introduced and studied by Baskakov [6] and Kurbatov
1,u
C
[32] independently, see also [25]. The above definition of the Gohberg-
Baskakov-Sj¨ostrand class (Zd,Zd) is given by Sun [43] for any 1
p,u
C ≤
p and any weight matrix u. From the definition of the Gohberg-
Ba≤sk∞akov-Sj¨ostrand class (Zd,Zd), we have the following inclusion:
p,u
C
(2.10) (Zd,Zd) (Zd,Zd)
p,u p,u
B ⊂ C
for any 1 p and for any weight matrix u.
≤ ≤ ∞
Remark 2.3. The inclusions (2.7) and (2.10) become equalities for
p = , i.e.,
∞
(2.11) (Zd,Zd) = (Zd,Zd) = (Zd,Zd) =: (Zd,Zd)
,u ,u ,u u
B∞ C∞ S∞ J
The class (Zd,Zd) of infinite matrices is usually known as the Jaffard
u
class, [6, 1J3, 23, 25, 29, 41, 43]. The Jaffard class (Zd,Zd) with poly-
u
J
nomial weight u := ((1+ i j )α) was introducedby Jaffard[29] in
i,j Zd
considering wavelets on a|n−op|en do∈main. The Jaffard class (Zd,Zd)
u
J
with weight matrices u := (v(i j)) associated with positive func-
i,j Zd
tions v on Rd was introduced b−y Bask∈akov [6] independently, and later
applied nontrivially in the study of localization of frames [25], adaptive
computation [13], and nonuniform sampling [42].
For the class of infinite matrices, we have the following propo-
p,u
B
sition.
Proposition 2.4. Let α C, 1 p , u be a weight matrix, and
∈ ≤ ≤ ∞
let A := ((a(i,j)) and B := (b(i,j)) belong to . Then
i,j Zd i,j Zd p,u
∈ ∈ B
(i) A+B A + B .
k kBp,u ≤ k kBp,u k kBp,u
(ii) αA = α A .
k kBp,u | |k kBp,u
(iii) A = A where A := a(j,i) is the conjugate
tkran∗kspBop,sue ofkthkeBmp,uatrix A. ∗ (cid:0) (cid:1)i,j∈Zd
(iv) A B if A B , i.e., a(i,j) b(i,j) for all
ik,jkBpZ,ud.≤ k kBp,u | | ≤ | | | | ≤ | |
∈
All conclusions in the above proposition follow directly from (2.2)
and (2.3). From the conclusions (i) and (ii) of the above proposition,
we see that is a norm on the class of infinite matrices. The
k·kBp,u Bp,u
properties in the conclusion (iv) is usually known as the solidness of
the matrix norm .
k·kBp,u
WIENER’S LEMMA FOR INFINITE MATRICES II 9
3. Algebraic properties
In this section, we establish some algebraic properties for the class
of infinite matrices and give a proof of Theorem 1.1.
p,u
B
Let us first recall the concept of a p-submultiplicative weight matrix
u [25, 43, 44]. For 1 p , a weight matrix u := (u(i,j)) is
i,j Zd
≤ ≤ ∞ ∈
said to p-submultiplicative if there exists another weight matrix v :=
(v(i,j)) such that
i,j Zd
∈
(3.1) v(i,j) 1 for all i,j Zd,
≥ ∈
(3.2) u(i,j) u(i,k)v(k,j)+v(i,k)u(k,j) for all i,j,k Zd,
≤ ∈
and
(3.3) C (v,u) := sup v(i,j)(u(i,j)) 1 < .
p −
(cid:13)(cid:13)(cid:16)|i−j|∞≥|k|∞(cid:0) (cid:1)(cid:17)(cid:13)(cid:13)p/(p−1) ∞
For p = 1, we simp(cid:13)ly say that a weight matrix is(cid:13)submultiplicative
instead of 1-submultiplicative. We call the weight matrix v satisfying
(3.1),(3.2)and(3.3)acompanionweightmatrixofthep-submultiplicative
weight matrix u. Denote by C(u) the set of all companion weights of a
p-submultiplicative weight matrix u, and define the p-submultiplicative
bound M (u) by
p
(3.4) M (u) := inf C (v,u).
p p
v C(u)
∈
One may verify that C(u) is a convex set and the infinimum of C (v,u)
p
in the set C(u) can be attained for some companion weight matrix v.
So from now on, except stated explicitly, we always assume that the
companion weight v of a p-submultiplicative weight matrix u is the one
satisfying
(3.5) M (u) = C (v,u).
p p
Remark 3.1. From the definitions of p-submultiplicative weight ma-
trices on Zd Zd, we have the following:
×
(i) A p-submultiplicative weight matrix is q-submultiplicative for
all 1 q p.
≤ ≤
(ii) A necessary condition for a weight matrix u := (u(i,j)) to
i,j Zd
∈
be p-submultiplicative is u(i,j) Cu(i,k)u(k,j) for all i,j,k
Zd and for some positive consta≤nt C. When p = 1, the abov∈e
necessary condition is also a sufficient condition [25].
(iii) Let 1 p ,δ (0,1), and let α be a number with the
≤ ≤ ∞ ∈
property that α > d d/p if 1 < p , and α 0 if
− ≤ ∞ ≥
p = 1. Then the Toeplitz matrices p := (1+ i j )α
α (cid:0) | − |∞ (cid:1)i,j∈Zd
10 QIYUSUN
generated by the polynomial weight (1 + x )α, and e :=
δ
exp( i j δ ) generated by the sub-e|x|p∞onential weight
| − | i,j Zd
e(cid:0)xp( x δ ), a∞re (cid:1)p-s∈ubmultiplicative [43].
| |
∞
Nowwestatethemainresultofthissection, anextensionofTheorem
1.1.
Theorem 3.2. Let 1 p,q , u be a p-submultiplicative weight
≤ ≤ ∞
matrix with the p-submultiplicative bound M (u), and let w be a discrete
p
A -weight with the A -bound A (w). Then the following statements
q q q
hold.
(i) If v is a companion weight matrix of the p-submultiplicative
weight matrix u, then
(3.6) AB 22/p5(d 1)/p A B + A B
−
k kBp,u ≤ k kBp,uk kB1,v k kB1,vk kBp,u
(cid:0) (cid:1)
for all A,B .
p,u
∈ B
(ii) is (and hence is also) an algebra. Moreover
p,u
B B
(3.7)
AB 21+2/p5(d 1)/pM (u) A B for all A,B .
k kBp,u ≤ − p k kBp,uk kBp,u ∈ Bp,u
(iii) is a subalgebra of . Moreover
p,u
B B
(3.8) A M (u) A for all A .
k kB ≤ p k kBp,u ∈ Bp,u
(iv) is (and hence is also) a subalgebra of (ℓq ). Moreover
Bp,u B B w
(3.9) Ac 22d3d/q(A (w))1/qM (u) A c
k kq,w ≤ q p k kBp,uk kq,w
for all A and c ℓq.
∈ Bp,u ∈ w
Before we give the proof of the above theorem, let us next have some
remarks on the unital Banach algebra property of the algebra , on
p,u
B
the equality of spectral radii in the algebras and , and on the
p,u 1,v
B B
inclusion (ℓq ).
Bp,u ⊂ B w
Remark 3.3. For 1 p and a p-submultiplicative weight matrix
≤ ≤ ∞
u := (u(i,j)) , following the standard procedure [19, 35] we define
i,j Zd
∈
A := sup AB for A . Then
k k′Bp,u kBkBp,u=1k kBp,u ∈ Bp,u
(3.10) AB A B for all A,B .
′ ′ ′ p,u
k k p,u ≤ k k p,uk k p,u ∈ B
B B B
If the weight matrix u further satisfies
(3.11) M := supu(i,i) < ,
∞
i Zd
∈
then the identity matrix I belongs to , and the norms and
Bp,u k·kBp,u
on are equivalent to each other, because
′ p,u
k·k p,u B
B
M 1 A A 21+2/p5(d 1)/pM (u) A for all A
− k kBp,u ≤ k k′Bp,u ≤ − p k kBp,u ∈ Bp,u