Let \(\phi \in L^2(\mathbb{R}^d)\) be a fixed normalized window function. Then the Gabor transform of a function \(f\in L^2(\mathbb{R}^d)\) with respect to \(\phi \) is defined as \((V_{\phi}f)(q,p)=\left\langle f,\phi _{q,p}\right\rangle _d\) where \(\phi _{q,p}(x)=\phi (x-q)e^{2\pi ipx}\). It is known that \((V_{\phi }f)(q,p)={\mathcal F}\left\{ f(.)\overline{\phi (.-q)}\right\} (p),\) where \({\mathcal F}\) is the Fourier transform of \(f(.)\overline{\phi (.-q)}.\) Consider the unitary operators \[ \begin{aligned} U_{1} &=I\otimes {\mathcal F}^{-1}:L^2(\mathbb{R}^{2d},dq\,dp)\\ &=L^2(\mathbb{R}^d,dq)\otimes L^2(\mathbb{R}^d,dp)\rightarrow L^2(\mathbb{R} ^d,dq)\otimes L^2(\mathbb{R}^d,dp)\end{aligned} \] and \[ U_2:L^2(\mathbb{R}^d,dq)\otimes L^2(\mathbb{R}^d,dp)\rightarrow L^2(\mathbb{R}^d,dq)\otimes L^2(\mathbb{R}^d,dp) \] given by \(U_2F(q,p)=F(p-q,p).\) The space \(V_{\phi }(L^2(\mathbb{R} ^d)),\) i.e. the images of \(L^2(\mathbb{R}^d)\) under the Gabor transform, will be called the space of Gabor transforms (Gabor space). These spaces are also called model spaces. The integral operator \[ (P_{\phi }F)(q,p)=\int\limits_{\mathbb{R}^{2d}}F(s,r)K_{s,r}(q,p)ds\,dr \] is the orthogonal projection onto the Gabor space, where \(K_{q,p}(s,r)=\left\langle\phi_{q,p},\phi _{r,s}\right\rangle _d\) is the kernel in the Gabor space with \[ F(q,p)=\left\langle F,K_{q,p}\right\rangle _{2d}=\int\limits_{\mathbb{R} ^{2d}}F(s,r)K_{s,r}(q,p)ds\,dr,\text{ }F\in V_{\phi}(L^2(\mathbb{R}^d)). \] In Section 2, the authors first characterize the Gabor space \(V_{\phi }(L^2(\mathbb{R} ^d))\) and prove the following theorem.
Theorem 1. The unitary operator \(U=U_2U_{1}\) gives an isometrical isomorphism of \(L^2(\mathbb{R} ^{2d},dq\,dp)\) onto itself under which
(i) the Gabor space \(V_{\phi }(L^2(\mathbb{R}^d))\) is mapped onto \(L_{\phi }\otimes L^2(\mathbb{R}^d,dp),\) where \(L_{\phi }\) is the one-dimensional subspace of \(L^2(\mathbb{R}^d,dq)\) generated by the function \(\ell _{\phi }(q)=\overline{\phi (q)};\)
(ii) the projection \(P_{\phi }\) is unitarily equivalent to \(UP_{\phi }U^{-1}=Q_{\phi }\otimes I,\) where \(Q_{\phi }\) is the one-dimensional projection of \(L^2(\mathbb{R} ^d,dq)\) onto \(L_{\phi }\) given as \[ (Q_{\phi }F)(q)=\overline{\phi (q)}\int\limits_{\mathbb{R}^d}F(r)\phi (r)dr. \] Denote by \(\Omega _1\) and \(\Omega _2=U_2(\Omega _1)\) the images of the space \(V_{\phi }(L^2(\mathbb{R}^d))\) under the mapping \(U_{1}\) and the set of all functions of the form \(F(q,p)=f(p)\overline{\phi (q)}\) respectively\(.\) Consider the isometrical imbedding \(R\phi:L^2(\mathbb{R}^d)\rightarrow L^2(\mathbb{R}^d)\otimes L^2(\mathbb{R}^d),\) \((R\phi f)(q,p)=f(p)\ell _{\phi }(q).\)
In this section the authors also prove the following theorem and give some corollaries.
Theorem 4. The operator \(R=R_{\phi }^{\ast }U\) maps the space \(L^2(\mathbb{R}^{2d},dq\,dp)\) onto \((L^2\mathbb{R}^d,dq),\) and the restriction \(R\mid_{V_{\phi }(L^2(\mathbb{R}^d))}:V_{\phi }(L^2(\mathbb{R}^d))\rightarrow L^2(\mathbb{R}^d)\) is an isometrical isomorphism given by \[ (RF)(\xi )=\int_{\mathbb{R}^{2d}}F(q,p)\phi (\xi -q)e^{2\pi ip\xi }dq\,dp. \tag{1} \] The adjoint operator \(R^{\ast }=U^{\ast }R_{\phi }:L^2(\mathbb{R}^d)\rightarrow V_{\phi }(L^2(\mathbb{R}^d))\) is an isometrical isomorphism of the space \(L^2(\mathbb{R}^d,dq)\) onto \(V_{\phi }(L^2(\mathbb{R} ^d))\subset L^2(\mathbb{R}^{2d},dq\,dp)\) given by \[ (R^{\ast }f)(q,p)=\int_{\mathbb{R}^d}f(\xi )\overline{\phi (\xi -q)}e^{-2\pi ip\xi}d\xi . \tag{2} \] Let \(S_{0}(\mathbb{R}^d)\) be the Feichtinger algebra and \(\phi \in S_{0}(\mathbb{R}^d)\). The Gabor-Toeplitz localization operator is defined as a map of \(L^2(\mathbb{R}^d)\) to \(L^2(\mathbb{R}^d)\) given by \[ (A_{\alpha }f)(x)=\int_{\mathbb{R}^{2d}}a(q,p)(V_{\phi }f)(q,p)\phi_{q,p}(x)dq\,dp, \] where \(a\) is assumed to be a bounded function defined on the phase space. \( a(q,p)\) is called the symbol of \(A_{\alpha }\) or its Gabor multiplier.
In Section 3, using the representation of the space \(V_{\phi }(L^2(\mathbb{R}^d))\), the authors prove the following important theorem and find some consequences of it.
Theorem 7. Let \(a=a(q)\) be a bounded measurable function. Then the Gabor-Toeplitz localization operator \(A_{\alpha }\) acting on \(V_{\phi}(L^2(\mathbb{R}^d))\) is unitarily equivalent to the multiplication operator \(\gamma _{a}I=RA_{\alpha }R^{\ast }\) acting on \(L^2(\mathbb{R}^d)\) where \(R\) and \(R^{\ast }\) are operators given by \((1)\) and \((2),\) respectively, and the function \(\gamma_{a}\) has the form \[ \gamma _{a}(r)=\int_{\mathbb{R}^d}a(r-s)\left| \ell _{\phi }(s)\right|^2ds,\;r\in \mathbb{R}^d. \]