On the relations of general homogeneous spaces and the two-wavelet localization operators

Abstract

In this note, the two-wavelet localization operator for square integrable representation of a general homogeneous space is defined. Then among other things, the boundedness properties of this operator is investigated. In particular, it is shown that it is in the Schatten p-class.

1 Introduction and preliminaries

The name of localization operator goes back to 1988, when I. Daubechies first used these operators as a mathematical tool to localize a signal on the time frequency plane. Recently, localization operators have been a subject of study in quantum mechanics; in PDE and signal analysis (see [1, 3, 10]). Wong [12] has studied the localization operators in the setting of homogeneous spaces with a G-invariant measure (see also [2]). The localization operators in the setting of homogeneous space with an admissible wavelet on general homogeneous space were studied in [11]. Also the two-wavelet localization operators on the homogeneous space G/H has been investigated in [4, 7, 8], where G/H was equipped with a relatively invariant measure. In this paper, we continue our investigation on the localization operators with two admissible wavelets in a completely different and more general approach, by considering a strongly quasi invariant measure on the homogeneous space G/H. The purpose of this paper is to explore and develop certain results concerning two wavelet localization operators in the setting of general homogeneous spaces. At this point for reader convenince and fixing our notations, we recall some remarkable pointers of Radon measures on homogeneous spaces.

Let G be a locally compact group and H be a closed subgroup of G. Consider G/H as a homogeneous space equipped with a strongly quasi invariant measure $\nu$. Note that G acts from the left on G/H in a natural way. We recall that a non-zero positive Radon measure $\nu$ is strongly quasi invariant if there is a continuous function $\kappa :G\times G/H \rightarrow (0,\infty )$ such that

$$\begin{aligned} d\nu _g(kH)=\kappa (g,kH) d\nu (kH), \end{aligned}$$

for all $g, k \in G$. If the function $\kappa (g,\cdot )$ reduces to constant, then $\nu$ is called relatively invariant under G. It is well known that any homogeneous space G/H possesses a strongly quasi invariant measure and all such measures are constructed by a rho-function $\rho$. We recall that a rho-function for the pair (G, H) is defined to be a continuous function $\rho :G\rightarrow (0,\infty )$ which satisfies

$$\begin{aligned} \rho (gh)=\frac{\Delta _H(h)}{\Delta _G(h)}\rho (g)\quad (g \in G ,h \in H), \end{aligned}$$

(1.1)

where $\Delta _G, \Delta _H$ are the modular functions on G and H, respectively (see [6]). For any pair (G,H) and each rho-function $\rho$, there is a strongly quasi invariant measure $\nu$ on G/H such that

$$\begin{aligned} d\nu _g(kH)=\kappa (g,kH) d\nu (kH)\quad (g,k \in G), \end{aligned}$$

in which

$$\begin{aligned} \kappa (g,kH) = \frac{\rho (gk)}{\rho (k)}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \kappa ({g_1}{g_2},x)= & {} \kappa ({g_1},{g_2}x)\kappa (g_2,x), \nonumber \\ \kappa (e,x)= & {} 1. \end{aligned}$$

(1.2)

A positive Radon measure $\nu$ on G/H is said to be G-invariant if $\nu _{g} = \nu$ for all g$\in$ G. It is well known that G/H admits a G-invariant measure when H is compact. (for a detailed account of Homogeneous spaces, the reader is referred to [6, 7, 9]).

Now, we recall the basic concepts of continuous unitary representations and continuous wavelet transform on homogeneous spaces [5]. For a Hilbert space ${\mathcal {H}}$, let $U({\mathcal {H}})$ denote the group of unitary operators on ${\mathcal {H}}$ equipped with the strong operator topology.

A continuous map

$$\begin{aligned} \varrho : G/H\longrightarrow & {} U({\mathcal {H}})\\ gH\longrightarrow & {} \langle \varrho (gH) x,y\rangle \end{aligned}$$

that $\varrho (gkH)= \varrho (gH) \varrho (kH) , \varrho (g^{-1}H) = \varrho (gH)^*$ for each $x, y \in {{\mathcal {H}}}$ and $g,k\in G,$ is called a continuous unitary representation of homogeneous space G/H. Moreover a closed subspace M of ${\mathcal {H}}$ is said to be invariant with respect to $\varrho$ if $\varrho (gH)M \subseteq M$, for all $g\in G$. A continuous unitary representation $\varrho$ is said to be irreducible if the only invariant subspaces of ${\mathcal {H}}$ are $\{0\}$ and ${\mathcal {H}}$.

Note that, there is a significant relationship between the representations of homogeneous space G/H and representations of group G. That is, if $\varrho$ is a representation of G/H, then it defines a representation $\sigma$ of G in which the subgroup H is considered to be contained in the kernel of $\sigma$. Conversely, any representation $\sigma$ of G which is trivial on H induces a representation $\varrho$ of G/H by letting $\varrho (gH) = \sigma (g)$.

Let $\varrho$ be a unitary irreducible representation of G/H on a Hilbert space ${\mathcal {H}}$. A vector $\eta \in {\mathcal {H}}$ is said to be admissible if

$$\begin{aligned} \int _{G/H} \kappa (g^{-1},gH) |\langle \eta , \varrho (gH) \eta \rangle | ^{2} d\nu (gH) < \infty . \end{aligned}$$

(1.3)

An admissible vector $\eta \in {\mathcal {H}}$ is called admissible wavelet if $\Vert \eta \Vert =1.$

Note that (1.3) is well-defined [11].

For a representation $\varrho$ of G/H on Hilbert space ${\mathcal {H}}$ and an admissible wavelet $\eta$ for $\varrho$, the continuous wavelet transform associated with the admissible wavelet $\eta$ is defined as the linear operator $\Omega _\eta$ from ${\mathcal {H}}$ into the space of continuous functions of G/H,

$$\begin{aligned} \Omega _{\eta } \Psi (gH) = \left( {\frac{\kappa (g^{-1},gH)}{\chi _\eta }}\right) ^{\frac{1}{2}} \langle \Psi , \varrho (gH) \eta \rangle , \end{aligned}$$

for all $\Psi \in {\mathcal {H}} , g\in G$ where

$$\begin{aligned} \chi _\eta := \int _{G/H} \kappa (g^{-1},gH)\ \ |\langle \eta , \varrho (gH) \eta \rangle | ^{2} d\nu (gH), \end{aligned}$$

(1.4)

is called the wavelet constant associated to $\eta$ in ${\mathcal {H}}$.

In the following theorem, we collect some properties of a square integrable representation of homogeneous G/H on a Hilbert space ${\mathcal {H}}$ with two admissible wavelets that are needed in the sequel.

Theorem 1.1

Let $\varrho$ be a square integrable representation of G/H on the Hilbert space ${\mathcal {H}}$ and $\eta , \xi$ be two admissible wavelets for $\varrho$. Then the following holds.

• (i) For all x, y in ${\mathcal {H}}$

  $$\begin{aligned} \int _{G/H} \frac{\kappa (g^{-1} , gH)}{\sqrt{\chi _\eta \chi _\xi }} \langle x,\varrho (gH)\eta \rangle \langle \varrho (gH)\xi ,y\rangle d\nu (gH) = \chi _{\eta ,\xi }\langle x,y\rangle , \end{aligned}$$

  in which

  $$\begin{aligned} \chi _{\eta ,\xi } = \int _{G/H} \frac{\kappa (g^{-1} , gH)}{\sqrt{\chi _\eta \chi _\xi }} \langle \eta ,\varrho (gH)\eta \rangle \langle \varrho (gH)\xi ,\eta \rangle d\nu (gH), \end{aligned}$$

• (ii) There exists a unique self adjoint positive operator C on ${\mathcal {H}}$ such that:

  • (a) The set of all admissible wavelets coincide with the domain of C.

  • (b) For all x, y in ${\mathcal {H}},$

    $$\begin{aligned} \int _{G/H} \frac{\kappa (g^{-1} , gH)}{\sqrt{\chi _\eta \chi _\xi }} \langle x,\varrho (gH)\eta \rangle \langle \varrho (gH)\xi ,y\rangle d\nu (gH) = \langle C\eta , C\xi \rangle \langle x,y\rangle \end{aligned}$$

Proof

The proof is the same as Theorem 2.1 of [11]. $\square$

The $\chi _{\eta ,\xi }$ in part(i) is called the two-wavelet constant.

In the rest of the paper, we aim to study the two-wavelet localization operators on a homogeneous space G/H equipped with a strongly quasi invariant measure which are related to a continuous wavelet transform. In Sect. 2, a two-wavelet localization operator which is related to continuous wavelet transform (C.W.T) is defined. In this setting, the boundedness properties of two-wavelet localization operators is investigated. In Sect. 3, we show that the two-wavelet localization operators are in p-Schatten class.

2 Two-wavelet localization operator

Throughout this section G is a locally compact group and H is a compact subgroup of G. Consider G/H as a homogeneous space equipped with a strongly quasi invariant measure $\nu$ which arises from a rho- function $\rho .$ Let ${\mathcal {H}}$ be a separable Hilbert space, $\varrho$ a square integrable representation of G/H on ${\mathcal {H}}$ and $\eta ,\xi$ are admissible wavelets for $\varrho$. In this section, first, we define a two-localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ which is related to the C.W.T $\Omega _\eta$ for each $\Theta \in L^{p}(G/H)$, $1\le p\le \infty$ and then we investigate the boundedness properties of this operator. To do so, we need the following definition.

Definition 2.1

Let ${\mathcal {H}}$ be a Hilbert space and $\varrho$ a square integrable representation of G/H on ${\mathcal {H}}$ with two admissible wavelets $\eta ,\xi$. A two-wavelet localization operator is a linear operator $\Upsilon _{\Theta ,\eta ,\xi }$ on ${\mathcal {H}}$ as

$$\begin{aligned} \langle \Upsilon _{\Theta ,\eta ,\xi } x,y\rangle= & {} \int _{G/H} \frac{\kappa (g^{-1},gH)}{\chi _{\eta ,\xi }} \Theta (gH)\nonumber \\&\langle x,\varrho (gH)\eta \rangle \langle \varrho (gH)\xi , y \rangle d\nu (gH), \end{aligned}$$

(2.1)

for all $\Theta \in L^{p}(G/H)$ and $x,y \in {\mathcal {H}}$, where $\chi _{\eta ,\xi }$ is the two-wavelet constant.

Now we show that the two-wavelet localization operator is bounded for $\Theta \in L^{p}(G/H)$, $1\le p\le \infty .$

The following proposition shows that $\Upsilon _{\Theta ,\eta ,\xi }$ is bounded for $\Theta \in L^{\infty }(G/H)$.

Proposition 2.2

By considering the notations above, let $\Theta \in L^{\infty }(G/H);$ Then $\Upsilon _{\Theta ,\eta ,\xi }$ is a bounded linear operator and

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le \frac{{(\chi {_\eta }\chi _{\xi }})^{\frac{1}{2}} }{|\chi _{\eta ,\xi }|}\Vert \Theta \Vert _{\infty }, \end{aligned}$$

in which $\eta$ and $\xi$ are two admissible wavelets.

Proof

To do this, using the reconstruction formula and the Schwartz inequality, we get,

$$\begin{aligned} \begin{aligned}&\vert \langle \Upsilon _{\Theta ,\eta ,\xi } x,y\rangle \vert \\&\quad \le \frac{1}{|\chi _{\eta ,\xi }|} \int _{G/H} \kappa (g^{-1} , gH) |\Theta (gH)| |\langle x, \varrho (gH)\eta \rangle ||\langle \varrho (gH)\xi ,y\rangle | d\nu (gH)\\&\quad \le \frac{1}{|\chi _{\eta ,\xi }|}\ {\Vert \Theta \Vert }_\infty \int _{G/H} \kappa (g^{-1},gH){|\langle x,\varrho (gH)\eta \rangle |}^2 d\nu (gH))^{\frac{1}{2}} \\&\qquad \left( \int _{G/H} \kappa (g^{-1},gH){|{\langle \varrho (gH)\xi ,y\rangle |}^2 }d\nu (gH)\right) ^{\frac{1}{2}}\\&\quad \le \frac{(\chi _\eta \chi _\xi )^\frac{1}{2}}{|\chi _{\eta ,\xi }|} \Vert \Theta \Vert _\infty \left( \int _{G/H}{|\Omega _{\eta }(x)|}^2d\nu \right) ^{\frac{1}{2}} \left( \int _{G/H}{|\overline{\Omega _{\eta }(y})|}^2 d\nu (gH)\right) ^{\frac{1}{2}} \\&\quad \le \frac{{(\chi _\eta {\chi _\xi })}^\frac{1}{2}}{|\chi _{\eta ,\xi }|}{\Vert \Theta \Vert }_\infty \Vert x\Vert \Vert y\Vert , \end{aligned}. \end{aligned}$$

So for $\Theta \in L^{\infty }(G/H)$,

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le \frac{{(\chi _\eta {\chi _\xi })}^\frac{1}{2}}{|\chi _{\eta ,\xi }|} {\Vert \Theta \Vert }_\infty . \end{aligned}$$

$\square$

As mentioned earlier, G/H admits such a G-invariant measure when H is compact. Now we show that $\Upsilon _{\Theta ,\eta ,\xi }$ is bounded for $\Theta \in L^{1}(G/H)$ in which G/H is equipped with a G-invariant measure $\nu ^\prime$.

Proposition 2.3

With the above notations, let $\Theta \in L^{p}(G/H), 1\le p< \infty$. Then $\Upsilon _{\Theta ,\eta ,\xi }$ is a bounded linear operator and

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}\Vert \Theta \Vert _p }. \end{aligned}$$

(2.2)

Proof

The proof is shown in two steps.

In the first step we show that if $\Theta \in L^1(G/H)$ then $\Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le {\frac{\rho (e)}{|\chi _{\eta ,\xi }|}}\Vert \Theta \Vert _1$.

In the second step we show that if $\Theta \in L^{p}(G/H)$, $1< p< \infty .$ then

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}\Vert \Theta \Vert _p }. \end{aligned}$$

(2.3)

Let $\Theta \in L^1(G/H)$. Consider G/H with a G-invariant measure ${\nu }^\prime$ which arises from rho-function $\rho ^\prime \equiv 1$ [6]. Note that since H is compact, G/H admits such a G-invariant measure $\nu ^{\prime }$ which is equivalent to $\nu$, that is $\frac{d\nu }{d{\nu }^\prime } = \varphi$ in which $\varphi (gH) = {\frac{\rho (g)}{\rho ^{\prime }(g)}}$, where $\nu$ is the strongly quasi invariant measure which arises from rho-function $\rho$. Thus

$$\begin{aligned} \begin{aligned}&|\langle \Upsilon _{\Theta ,\eta ,\xi } x,y\rangle |\\&\quad \le \ \frac{1}{|\chi _{\eta ,\xi |}}\int _{G/H} \kappa (g^{-1} , gH) |\Theta (gH)| |\langle x, \varrho (gH)\eta \rangle ||\langle \varrho (gH)\xi ,y\rangle |d\nu (gH) \\&\quad \le \ \frac{1}{|\chi _\eta ,\xi |}\int _{G/H}\frac{\rho (e)}{\varphi (gH)}|\Theta (gH)| |\langle x, \varrho (gH)\eta \rangle ||\langle \varrho (gH)\xi ,y\rangle | \varphi (gH) {d{\nu }^\prime }(gH) \\&\quad \le \frac{1}{|\chi _{\eta ,\xi }|}\int _{G/H} \rho (e) \Vert \Vert x\Vert \Vert y\Vert |\Theta (gH)| {d{\nu }^\prime }(gH)\\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|} {{\Vert \Theta \Vert }_1 } \Vert \Vert x\Vert \Vert y\Vert , \end{aligned} \end{aligned}$$

where $\Vert \Theta \Vert _1 = \int _{G/H} |\Theta (gH)| {d{\nu }^\prime }(gH)$.

Now through the use of the Reisz Thorin Theorem, we shows that if $\Theta \in L^{p}(G/H) , 1< p< \infty$, then there exists a unique bounded linear operator $\Upsilon _{\Theta ,\eta ,\xi }: {\mathcal {H}}\longrightarrow {\mathcal {H}}$ such that

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert \le \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}\Vert \Theta \Vert _p }, \end{aligned}$$

where $\Upsilon _{\Theta ,\eta ,\xi }$ is given in (2.1) for a simple function $\Theta$ on G/H for which

$$\begin{aligned} \nu (\{ gH \in G/H; \Theta (gH)\ne 0\}) < \infty . \end{aligned}$$

(2.4)

Let $\Gamma : {\mathcal {H}}\longrightarrow L^{2}(R^n)$ be a unitary operator and $\Theta \in L^{1}(G/H).$ Then the linear operator ${\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } : L^{2}(R^n)\longrightarrow L^{2}(R^n)$ defined by

$$\begin{aligned} {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } = \Gamma \Upsilon _{\Theta ,\eta ,\xi }{\Gamma }^{-1}, \end{aligned}$$

is bounded and $\Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } \Vert \le {\frac{\rho (e)}{|\chi _{\eta ,\xi }|}}\Vert \Theta \Vert _1.$ If $\Theta \in L^{\infty }(G/H)$, then the linear operator ${\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }$ on $L^{2}(R^n)$ is bounded and $\Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } \Vert \le {\frac{{(\chi {_\eta }\chi _{\xi }})^{\frac{1}{2}} }{|\chi _\eta ,\xi |} }\Vert \Theta \Vert _{\infty }.$

Denote by S the set of all simple functions $\Theta$ on G/H which satisfy the condition (2.4). Let $g\in L^{2}(R^n)$ and $\Phi _g$ be a linear transformation from S into the set of all Lebesgue measurable function on $R^n$ defined as $\Phi _g(\Theta )={\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } (g)$. Then for all $\Theta \in L^{1}(G/H)$

$$\begin{aligned} \Vert \Phi _g(\Theta ) \Vert _2 =\Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi } (g)\Vert _2 \le \Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }\Vert \Vert g\Vert _2 \le {\frac{\rho (e)}{|\chi _{\eta ,\xi }|}}\Vert \Theta \Vert _1 \Vert g\Vert _2. \end{aligned}$$

Similarly for $\Theta \in L^{\infty }(G/H),$

$$\begin{aligned} \Vert \Phi _g(\Theta ) \Vert _2 \le \frac{{(\chi {_\eta }\chi _{\xi }})^{\frac{1}{2}} }{|\chi _{\eta ,\xi }|}\Vert \Theta \Vert _{\infty }\Vert g\Vert _2. \end{aligned}$$

Through the Riesz Thorin Interpolation Theorem we get,

$$\begin{aligned} \Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }(g)\Vert _2 = \Vert \Phi _g(\Theta ) \Vert _2 \le {\frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|}} (\chi _\eta {\chi _\xi })^{\frac{1}{2}(1-\frac{1}{p})}\Vert \Theta \Vert _p{\Vert g\Vert _2}. \end{aligned}$$

So,

$$\begin{aligned} \Vert {\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }\Vert \le {\frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|}} (\chi _\eta {\chi _\xi })^{\frac{1}{2}(1-\frac{1}{p})} \Vert \Theta \Vert _p. \end{aligned}$$

for each $\Theta \in S.$

Now, let $\Theta \in L^{p}(G/H)$, for $1<p<\infty .$ Then there exists a sequence $\{\Theta _k\}^{\infty }_{k=1}$ of functions in S such that $\Theta _k$ is convergent to $\Theta$ in $L^{p}(G/H)$ as $k\longrightarrow \infty .$ Also, $\{{\Upsilon ^\thicksim }_{{\ {\Theta }_k},\eta ,\xi }\}$ is a Cauchy sequence in $B( L^{2}({\mathbb {R}}^n))$. Indeed,

$$\begin{aligned} \Vert {\Upsilon ^\thicksim }_{ {\Theta _k},\eta ,\xi } - {\Upsilon ^\thicksim }_{ {\Theta _m},\eta ,\xi }\Vert \le {\frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|}} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}}\Vert \Theta _k - \Theta _m\Vert _p \longrightarrow 0. \end{aligned}$$

By the completeness of $B( L^{2}(R^n))$, there exists a bounded linear operator ${\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }$ on $L^{2}(R^n)$ such that ${\Upsilon ^\thicksim }_{{ \Theta _n},\eta ,\xi }$ converges to ${\Upsilon ^\thicksim }_{\Theta ,\eta ,\xi }$ in $B( L^{2}(R^n))$, in which

$$\begin{aligned} \Vert {\Upsilon ^\thicksim }_{ \Theta ,\eta ,\xi } \Vert \le {\frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|}} (\chi _\eta {\chi _\xi })^{\frac{1}{2}(1-\frac{1}{p})}\Vert \Theta \Vert _p. \end{aligned}$$

Thus the linear operator $\Upsilon _{\Theta ,\eta ,\xi }$is bounded,where $\Upsilon _{\Theta ,\eta ,\xi } = \Gamma ^{-1}{\Upsilon ^\thicksim }_{ \Theta ,\eta ,\xi }\Gamma$ and

$$\begin{aligned} \Vert \Upsilon _{ \Theta ,\eta ,\xi } \Vert \le \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-\frac{1}{p})}\Vert \Theta \Vert _p. \end{aligned}$$

For the proof of uniqueness, let $\Theta \in L^{p}(G/H), 1<p<\infty ,$ and suppose that $P_{\Theta ,\eta ,\xi }$ is another bounded linear operator satisfying (2.3). Let $\Xi : L^{p}(G/H)\longrightarrow B(\mathcal {H)}$ be the linear operator defined by

$$\begin{aligned} \Xi (\Theta ) =\Upsilon _{\Theta ,\eta ,\xi } - {P_ {\Theta ,\eta ,\xi }},\quad \Theta \in L^{p}(G/H). \end{aligned}$$

Then

$$\begin{aligned} \Vert \Xi (\Theta )\Vert \le 2 \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}} \Vert \Theta \Vert _p. \end{aligned}$$

Moreover $\Xi (\Theta )$ is equal to the zero operator on ${\mathcal {H}}$ for all $\Theta \in S$. Thus, $\Xi : L^{p}(G/H)\longrightarrow B(\mathcal {H)}$ is a bounded linear operator that is equal to zero on the dense subspace S of $L^{p}(G/H).$ Therefore $\Upsilon _{\Theta ,\eta ,\xi } = P_{\Theta ,\eta ,\xi }$ for all $\Theta \in L^{p}(G/H).$ $\square$

3 Compactness and Schatten p-class

Let ${\mathcal {H}}$ be a separable Hilbert space and $B_1$ the closed unit ball of ${\mathcal {H}}$. We recall that a linear operator T on ${\mathcal {H}}$ is said to be compact if $T(B_1)$ is compact in the norm topology of ${\mathcal {H}}$. It is well known that a linear operator T on ${\mathcal {H}}$ is compact if and only if $T(B_1)$ has compact closure in ${\mathcal {H}}$. Now Suppose $\{\alpha _n\}_{n\in {\mathbb {N}}}$ and $\{\beta _n\}_{n\in {\mathbb {N}}}$ are orthonormal sets in ${\mathcal {H}}$ and $\{\lambda _n\}$ is a sequence of complex numbers tending to 0. Let T be the linear operator on ${\mathcal {H}}$ defined by

$$\begin{aligned} T\eta = \Sigma _{n=1}^{\infty } \lambda _n\langle \eta , \alpha _n\rangle \beta _n, \quad \eta \in {\mathcal {H}} \end{aligned}$$

then T is compact. Note that $\{\lambda _n\}$ are eigenvalues of $\vert T \vert =(T^*T)^{\frac{1}{2}}$.

Given $0<p<\infty$, the Schatten p-class of ${\mathcal {H}}$, denoted by $S_p$, is the space of all compact operators T on ${\mathcal {H}}$ with its singular value sequence $\{\lambda _n\}$ belonging to $l^p$, the p-summable sequence space.

In the case, $1\le p<\infty$, $S_p$ is a Banach space with the norm

$$\begin{aligned} \Vert T\Vert _p= [\Sigma {_n}|\lambda _n |^p]^{\frac{1}{p}}. \end{aligned}$$

Two special cases are worth mentioning: $S_1$ is usually called the trace class and $S_2$ is usually called the Hilbert–Schmidt class.

For any $T\in S_1$ and any orthonormal basis $\{\alpha _k\}_{k=1}^{\infty }$ of ${\mathcal {H}},$ we write

$$\begin{aligned} tr(T) = \Sigma _{k=1}^{\infty }\langle T \alpha _{k},\alpha _{k}\rangle , \end{aligned}$$

and call this value, the trace of T. (for more details on Schatten p-classes see [12, 13]).

Now we show that the localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ is in the p-Schatten class.

Theorem 3.1

Based on the above notations, let $\Theta \in L^{p}(G/H), 1\le p\le \infty$. Then $\Upsilon _{\Theta ,\eta ,\xi }: {\mathcal {H}}\longrightarrow {\mathcal {H}}$ is in $S_p$ and

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert _{S_p }\le \frac{\rho (e)^{\frac{1}{p}}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{{\frac{1}{2}}(1-{\frac{1}{p})}}\Vert \Theta \Vert _p, \end{aligned}$$

where ${\Vert \Theta \Vert _p}^p = \int _{G/H}{|\Theta (gH)|}^pd{\nu ^\prime (gH)}$ and $\nu ^\prime$ is a G-invariant measure on G/H.

Proof

Let $\Theta \in L^{1}(G/H)$, $\{ \eta _k\}_{k=1}^\infty$ and $\{\xi _k\}_{k=1}^\infty$ be any orthonormal set of ${\mathcal {H}}$ and $\nu ^\prime$ be a G-invariant measure on G/H which arises from ${\rho ^\prime }(g)\equiv 1$, we get

$$\begin{aligned} \begin{aligned}&{\Sigma _{k=1}^\infty } |\langle \Upsilon _{\Theta ,\eta ,\xi } {\eta _k},\eta _k\rangle | \\&\quad \le \ \frac{1}{|\chi _{\eta ,\xi }|}{\Sigma _{k=1}^\infty } \int _{G/H} \kappa (g^{-1},gH) |\Theta (gH)| |\langle \eta _k, \varrho (gH)\eta \rangle | |\langle \varrho (gH)\xi ,\eta _k\rangle |d\nu (gH)\\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|} \int _{G/H} |\Theta (gH)| ({\Sigma _{k=1}^\infty }|\langle \eta _k, \varrho (gH)\eta \rangle |^2)^{\frac{1}{2}} ({\Sigma _{k=1}^\infty }|\langle \varrho (gH)\xi ,\eta _k\rangle |^2)^{\frac{1}{2}} d{\nu ^\prime } (gH)\\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|} \int _{G/H} |\Theta (gH)| d{\nu ^\prime } (gH)\\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|}\Vert \Theta \Vert _1 <\infty . \end{aligned} \end{aligned}$$

So $\Upsilon _{\Theta ,\eta ,\xi }\in S_1$. Now by Proposition 2.2, the two-wavelet localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ is a bounded linear operator for $\Theta \in L^{\infty }(G/H)$ and

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi }\Vert _{S_\infty }\le {\frac{(\chi _\eta \chi _\xi )^\frac{1}{2}}{|\chi _{\eta ,\xi }|}}\Vert \Theta \Vert _{\infty }. \end{aligned}$$

Then Reisz Torin Interpolation Theorem [13] implies that the localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ for $1\le p\le \infty$ is in $S_p$. $\square$

Here, we obtain a formula for the trace class of the localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ where $\Theta \in L^{1}(G/H).$

Theorem 3.2

Let $\Theta \in L^{1}(G/H)$ be a positive function. Then $tr(\Upsilon _{\Theta ,\eta ,\xi })= \frac{\rho (e)}{\chi _{\eta ,\xi }} \langle \xi ,\eta \rangle \Vert \Theta \Vert _1$ in which $\nu ^\prime$ is a G-invariant measure on G/H.

Proof

First note that, since H is compact, G/H admits a G-invariant measure $\nu ^ \prime$. Also we equipped G/H with a strongly quasi invariant measure $\nu$ which arises from a rho-function $\rho$. These two measures are related as follows, $\frac{d\nu }{d{\nu }^\prime } = \varphi$, where $\varphi (gH) = \rho (g)$. Now, let $\{ \eta _k ; k=1,2,\ldots \}$ be an orthonormal basis for ${\mathcal {H}}$. Then by using Fubini’s theorem, Parseval’s identity, for $\Theta \in L^{1}(G/H)$ we get

$$\begin{aligned} tr(\Upsilon _{\Theta ,\eta ,\xi })= & {} {\Sigma _{k=1}^\infty } (\Upsilon _{\Theta ,\eta ,\xi }{\eta _k}, {\eta _k}) \\= & {} \frac{1}{\chi _{\eta ,\xi }}\Sigma _{k=1}^\infty {\int _{G/H}}\kappa ({g^{-1}},gH) \Theta (gH)\langle \eta _k,\varrho (gH)\eta \rangle \\&\langle \varrho (gH)\xi ,\eta _k\rangle \rho (g)d{\nu }^{\prime }(gH) \\= & {} \frac{\rho (e)}{\chi _{\eta ,\xi }}\int _{G/H} \langle \varrho (gH)\xi ,\varrho (gH)\eta \rangle {\Theta (gH)} d{\nu }^\prime (gH)\\= & {} {\frac{\rho (e)}{\chi _\eta ,\xi }}\langle \xi ,\eta \rangle \Vert \Theta \Vert _1. \end{aligned}$$

$\square$

In the next theorem, we show that the localization operator $\Upsilon _{\Theta ,\eta ,\xi }$ is compact.

Theorem 3.3

Let $\Theta \in L^{p}(G/H) , 1\le p <\infty$. Then $\Upsilon _{\Theta ,\eta ,\xi }: {\mathcal {H}}\longrightarrow {\mathcal {H}}$ is a compact operator.

Proof

Let $\Theta \in L^{p}(G/H)$ with a G-invariant measure $\nu ^\prime$. There exists $\Theta _n \in C{_c}(G/H)$ such that $\Theta {_n}-\Theta \longrightarrow 0$. Let $\{ \eta _k ; k=1,2,\ldots \}$ be an orthonormal basis for ${\mathcal {H}}$. Then by Fubini’s Theorem and the Schwartz inequality, we have

$$\begin{aligned} \begin{aligned}&{\Sigma _{k=1}^\infty } |\langle \Upsilon _{{\Theta _n},\eta ,\xi }{\eta _k},\Upsilon _ {{\Theta _n},\eta ,\xi }{\eta _k}\rangle \\&\quad \le \ \frac{1}{|\chi _{\eta ,\xi }|} \Sigma _{k=1}^\infty {\int _{G/H}\kappa (g^{-1},gH) | \Theta _n}(gH)|\langle \eta _k,\varrho (gH)\eta \rangle | \\&\qquad |\langle \varrho (gH)\xi , \Upsilon _{{\Theta _n},\eta ,\xi } {\eta _k} \rangle | \rho (g)d{\nu }^{\prime }(gH) \\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|}\int _{G/H} |\Theta _n(gH)| (\Sigma _{k=1}^\infty {|\langle \eta _k,\varrho (gH)\eta \rangle |}^2)^\frac{1}{2} \\&\qquad (\Sigma _{k=1}^\infty {|\langle \Upsilon _{{\Theta _n},\eta ,\xi }^*\varrho (gH)\xi ,\eta _k\rangle |}^2)^\frac{1}{2} d{\nu }^\prime (gH)\\&\quad \le \ \frac{\rho (e)}{|\chi _{\eta ,\xi }|} \Vert \Upsilon _{\Theta _n,\eta ,\xi }^* \Vert \Theta _n\Vert _1 <\infty . \end{aligned} \end{aligned}$$

Thus $\Upsilon _{{\Theta _n},\eta ,\xi }$ is in $S_2$ and it implies that $\Upsilon _{{\Theta _n},\eta ,\xi }$ is compact. Since

$$\begin{aligned} \Vert \Upsilon _{\Theta ,\eta ,\xi } - \Upsilon _{{\Theta _n},\eta ,\xi }\Vert \le \frac{\rho (e)^\frac{1}{p}}{|\chi _{\eta ,\xi }|} (\chi _\eta {\chi _\xi })^{\frac{1}{2}(1-\frac{1}{p})} \Vert \Theta - \Theta _n \Vert _p. \end{aligned}$$

Then the two-wavelet localization operator is compact. $\square$