International Journal of Analysis and Applications
ISSN 2291-8639
Volume 8, Number 2 (2015), 100-103
http://www.etamaths.com

GELFAND TRIPLE ISOMORPHISMS FOR WEIGHTED BANACH SPACES

ON LOCALLY COMPACT GROUPS

S.S. PANDEY AND ASHISH KUMAR∗

Abstract. As in [1], we use the concept of wavelet transform on a locally compact group
G to construct weighted Banach spaces H1w(G), we being a submultiple weight function
on G. The main result of this paper provides an extension of a unitary mapping U from
H(G1) to H(G2) under suitable conditions to an isomorphism between the Gelfand triple
(H1w, H, H1∼w )(G1) and (H1w, H, H1∼w )(G2); where G1, G2 are any two locally compact group-
s, H a Hilbert space and H1∼w is the space of all continuous-conjugate linear functional on
H1w. This paper paves the way for the study of some other properties of Gelfand triples.

1. Introduction

I.M. Gelfand introduced a triple of abstract space consisting of a Frechet space F of test
functions continuously and densely embedded in Hilbert space H while H itself is continuously
and densely embedded in the dual space F′ of F (for details see [3]). Feichtinger and Kozek [2]
have studied a number of properties of Gelfand triple replacing the Frechet space F by a suit-
able Banach space. In particular, they have discussed in detail the extensions of isomorphisms
between L2-spaces on elementary locally compact abelian groups G to Gelfand triples of the
form (S0,L

2,S′0)(G) where S0(G) is the well known Feichtinger algebra, which has a number of
highly useful functional properties ([2], p. 237). Also, they have studied some important prop-
erties of the operator Gelfand triple (B, H, B′), where B is the Banach space of all bounded
linear operators from S′0(G) to S0(G) with respect to the operator norm ‖ ·‖op.

In the present paper, following Feichtinger and Gröchenig ([1], p. 309), we define Wavelet
transform Vgf of a function f with respect to g, both as elements of a Hilbert space H, on
a locally compact group G. Using these wavelet transforms, we construct a weighted Banach
spaces H1w(G) as in ([1], p. 317), where w is a submultiplicative weight function on G. We prove
that any unitary map U from H(G1) to H(G2) extends as an isomorphism from the Gelfand
triple (H1w,H,H1∼w )(G1) to (H1w,H,H1∼w )(G2) if and only if the restrictions of U and U∗ are
bounded linear operators from H1w(G1) to H1w(G2) respectively, where H1∼w (G1) is the Banach
space of all continuous-conjugate linear functionals on H1w(G).

This paper paves the way for the study of some properties associated with these Gelfand
triples.

2. Notations and Basic Concepts.

Let G be a locally compact group and dx the normalized Haar measure on it. We assume
that w : G → R+ is a submultiplicative weight function on G such that

w(x◦y) ≤ w(x) w(y)
for all x, y ∈ G

2010 Mathematics Subject Classification. 43A15, 43A32, 43B65 and 47A67.

Key words and phrases. Wavelet transform on locally compact groups; Unitary mappings and Gelfand triples.

c©2015 Authors retain the copyrights of their
papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

100



GELFAND TRIPLE ISOMORPHISMS 101

We suppose that all weight on G are symmetrical, i.e.,
w(x) = w(−x), ∀ x ∈ G.

We denote by Lpw(G), 1 ≤ p < ∞, the Banach space of functions on G with respect to the
norm

‖f‖p,w =
(∫

G

|f(x)|p wp(x) dx
)1/p

< ∞. (2.1)

In case p = ∞, we define the space L∞w (G) as the Banach space of all measurable functions
f on G such that

‖f‖∞,w = ess sup{|f(x)| w(x) : x ∈ G} < ∞. (2.2)

The conjugate space of Lpw(G) is the space L
p′

w−1
(G), where 1/p + 1/p′ = 1..

It is well known that L1w(G) is a commutative Banach algebra under convolution , which is
usually known as Beurling algebra, and we have the properties:

Lpw(G) ∗L
1
w(G) ⊆ L

p
w(G)

and

‖f ∗g‖≤‖f‖p,w ‖g‖1,w
for all f ∈ Lpw(G) and g ∈ L1w(G).

3. Wavelet Transform on G.

Let (π,H) be an irreducible continuous unitary representation of a locally compact group G
on a Hilbert space H(G). If f,g ∈H, then the wavelet transform of f with respect to g is given
by ([1], p. 317):

Vgf : x →〈π(x)g, f〉,
where

Vgf(x) =

∫
G

π(x) g(x)f̄(y) dy

and f̄(y) is the complex conjugate of f(y).

The representation π is called square integrable provided Vgf ∈ L1(G), ∀ g ∈ H. It is
known that if π is square-integrable, i.e., Vgg ∈ L2(G), then ∃ a unique, positive, self-adjoint
and densely defined operator A on H satisfying the following orthogonality relation ([1], pp.
309-310): ∫

G

Vg1f1(x) Vg2f2(x) dx = 〈Ag2, Ag1〉〈f1,f2〉

for all f1,f2 ∈H and g1,g2 ∈ dom A.
In case f1 = f2 = g1 = g2 = g ∈ dom A, f1 = f and ‖g1‖2 = 1, then we have

Vg f ∗Vg g = Vg f〈g,g〉
= Vg.

Now, on the line of Feichtinger and Gröchenig ([1], p.317),we define the set of analyzing vector
h1w(G) by

h1w(G) = {g : g ∈H,Vg g ∈ L
1
w(G)}.

Since π is irreducible ,h1w is a dense linear subspace of H. We suppose that h1w(G) is non-trivial
and g is a non-zero fixed element of h1w(G). we define H1w(G) by

H1w(G) = {f; f ∈H,Vg f ∈ L
1
w},

which is a Banach space under the norm

‖f|H1w‖ = ‖Vg f|L
1
w‖.



102 PANDEY AND KUMAR

As mention by Feichtinger and Gröchenig (loc. cit.),H1w(G) is a π-invariant Banach space dense
in H and the set {π(x)g,x ∈ G} is a total subset of H1w(G)

We denote by H1
∼

w (G) the Banach space of all continuous-conjugate linear functionals on
H1w(G). Hence H1

∼

w (G) is a π-invariant Banach space with the continuous dense embeddings

H1w ↪→H ↪→H
1∼

w

Which insure that

(H1w, H, H
1∼

w )

forms a Gabor triple (for detail see [3])

4. Extension of Unitary Gelfand Triple Isomorphisms

In a recent paper Feichtinger and Kozek ([2], pp.239-240) have shown that a unitary map-
ping U acting from L2(G1) to L

2(G2) extends to an isomorphism between the Gelfand triples
(S0, L

2, S′0)(G1) and (S0, L
2, S′0)(G2) if an only if the restrictions of U and U

∗ are bounded
linear operators between S0(G1)) and S0(G2), where S0(G1) and S0(G2) denote the Feichtinger
algebras on elementary locally compact abelian groups G1 and G2 respectively and S

′
0(G1)) and

S′0(G)1) their topological duals. Also, they have pointed out some applications of the above
isomorphism ([2], p.239).

In this section, on the lines of Feichtinger and Kozek, we study an extension of a unitary map-
ping U acting from H to H to an isomorphism between the Gelfand triples (H1w,H,H1∼w )(G1)
and (H1w,H,H1∼w )(G2), where G1, G2 are any two locally compact groups.

Precisely, we prove the following :

Theorem 4.1. If U is an unitary operator from H(G1) to H(G2), then it extends isomorphism
from the Gelfand triples where G1, G2 are any two locally compact groups (H1w,H,H1∼w )(G1)
and (H1w,H,H1∼w )(G2) if and only if there exists a positive constant c such that

‖Uf|H1w(G2)‖≤ c ‖f|H
1
w(G1)‖, ∀ f ∈H

1
w(G1) (4.1)

and

‖U∗f|H1w(G1)‖≤ c ‖f|H
1
w(G2)‖, ∀ f ∈H

1
w(G2) (4.2)

where U∗ is the adjoint operator of U.

Proof: The proof follows on the lines of Feichtinger and Kozek ([2], p. 240). But, since
our settings are different, it is necessary to give the proof.

Let us assume that (4.1) holds true. Then by virtue of the relation

〈Ūg,f〉 = 〈g,U∗f〉, ∀ g ∈H1∼w (G1) and f ∈H
1∼
w (G2),

we see that g →Ū g is a bounded linear mapping, which extends the unitary map U on H(G1).
Next, since H(G1) is boundedly dense in H1∼w (G1), Ū is a continuous and bounded linear map-
ping from H1∼w (G1) to H1∼w (G2). Also, Ū is unique and it coincides with U on H(G1). In the
same way it is clear that Ū∗ is a unique, continuous and bounded linear mapping from H1∼w (G2)
to H1∼w (G1), which coincides with U∗ on H(G2).

Thus we infer that Ū defines an isomorphism between H1∼w (G1) and H1∼w (G2) with respect
to their norm topologies.

Conversely, we suppose that U is a unitary operator, which extends as an isomorphism from
(H1w,H,H1∼w )(G1) and (H1w,H,H1∼w )(G2). Hence the restrictions of U and U∗ are bounded



GELFAND TRIPLE ISOMORPHISMS 103

linear mapping on the spaces H1w(G1) and H1w(G2) respectively.

This completes the proof of the theorem.

As a corollary of the above theorem, we show that the Gelfand triples isomorphisms holds
true provided there exists a bijection mapping V between H(G1) and H(G2) such that

〈f1 f2〉H(G1) = 〈V f1, V f2〉H(G2)
for all f1, f2 ∈H(G1).

As in [2], p.240), we prove the following :

Corollary: If V : H1w(G1) →H1w(G2) is an isomorphism, then it extends to a unitary Gelfand
triples isomorphism between (H1w,H,H1∼w )(G1) and (H1w,H,H1∼w )(G2) if and only if

〈f1, f2〉H(G1) = 〈V f1, V f2〉H(G2) (4.3)
for all f1, f2 ∈H1w(G1).

Proof: The isomorphism V as defined above is a unitary operator isomorphism from (H1w,H,H1∼w )(G1)
to (H1w,H,H1∼w )(G2) provided the condition (4.3) holds true.

Conversely, we suppose that the condition (4.3) holds. Then we have

‖f‖2H(G1) = 〈f, f〉H(G1)
= 〈V f, V f〉H(G2)
= ‖V f‖2H(G2), ∀ f ∈H

1
w(G1).

⇒ ‖f‖2H(G1) = ‖V f‖H(G2) say U,

⇒ V is extends to an isomorphism mapping, from H(G1) to H(G2).

Next, since V (H1w(G1)) = H1w(G1) is dense in H(G2) U has a dense range in H(G2).

⇒ U is an isomorphism from H(G1) to H(G2).

Hence, by the duality condition, U is an isomorphism between the Gelfand triples (H1w,H,H1∼w )(G1)
and (H1w,H,H1∼w )(G2)
This complete the proof the theorem

References

[1] H. G. Feichtinger and K. Gröchenig. Banach spaces related to integrable group representations and their
atomic decompositions I, Journal of Functional analysis, 86 (1989), 307-340.

[2] H. G. Feichtinger and W. Kozek. Quantization of TF lattice-invariant operators on elementary LCA groups.

In H.G. Feichtinger and T. Ströhmer, Editors, Gabor Analysis and Alorithms: Theory and Applications,
Birkhaüser, Boston, 1998, 233-266.

[3] I.M. Gelfand and N.J. Wilenkin. Genralized Functions, Vol. IV: Some Applications of Harmonic Analysis.
Rigged HIlbert Spaces. Academic Press, New York, 1964.

Department of Mathematics, R.D. University ,Jabalpur, India

∗Corresponding author