Matrix representation of Toeplitz operators on Newton spaces (Q6552083)

The Toeplitz operators are a class of bounded linear operators defined on Banach spaces. They play a significant role in various fields such as signal processing, control theory, and complex analysis. Here's a general definition of of Toeplitz operators:\N\NLet \(H=L^2(\mathbb{T})\) be Hilbert space where \(\mathbb{T}\) is a unit circle, and \(\phi \in L^2(\mathbb{T}) .\) Toeplitz operator is\N\[\NT_\phi(f)=\mathrm{Proj} (\phi f) \ \ \ (f \in H^2)\N\]\Nwhere \(\phi f\) is a multiplication operation and Proj is is the orthogonal projection of \(L^2(\mathbb{T})\) onto the Hardy space \(H^2.\)\N\NThe paper ''Matrix representation of Toeplitz operators on Newton spaces'' authored by Eungil Ko, Ji Eun Lee and Jongrak Lee, considered a special class of Toeplitz operator.\N\NThe Toeplitz operator defined on Newton space as\N\[\NT_\phi(f)=\mathrm{Proj} (\phi f) \ \ \ (f \in N^2(\mathbb{P}))\N\]\Nwhere\N\[\NN^2(\mathbb{P})= \left\{ f(z) = \sum_{n=0}^{\infty} a_n N_n(z) : \|f\|^2 = \sum_{n=0}^{\infty} |a_n|^2 < \infty \right\}\N\]\Nbe a Newton space as the closure of the set of polynomials in \(L^2(\mathbb{C},\mu).\)\N\NMoreover, this paper investigates the matrix representation of Toeplitz operators with respect to such an orthonormal basis on Newton space \( N^2(\mathbb{P}).\)\N\NThe most important results Theorem 2.11:\N\NTheorem 2.11. For the harmonic symbol \( \phi(z) = \sum_{i=0}^{\infty} a_i N_i + \sum_{i=1}^{\infty} a_{-i} N_i \), the matrix of \( T_\phi \) with respect to the orthonormal basis \( B = \{N_n\}_{n \geq 0} \) is given by\N\[\N[T_\phi]_B = \begin{pmatrix} a_0 b_0(0, 0) & a_{-1} b_1(1, 0) & a_{-2} b_2(2, 0) & a_{-3} b_3(3, 0) & \cdots \\\Na_1 b_1(1, 0) & \sum_{i=0}^{1} a_i b_1(i, 1) & \sum_{i=1}^{2} a_{-i} b_2(i, 1) & \sum_{i=2}^{3} a_{-i} b_3(i, 1) & \cdots \\\Na_2 b_2(2, 0) & \sum_{i=1}^{2} a_i b_2(i, 1) & \sum_{i=0}^{2} a_i b_2(i, 2) & \sum_{i=1}^{3} a_{-i} b_3(i, 2) & \cdots \\\Na_3 b_3(3, 0) & \sum_{i=2}^{3} a_i b_3(i, 1) & \sum_{i=1}^{3} a_i b_3(i, 2) & \sum_{i=0}^{3} a_i b_3(i, 3) & \cdots \\\N\vdots & \vdots & \vdots & \vdots & \ddots \\\N\end{pmatrix}\N\]\Nand the adjoint of the matrix of \( T_\phi \) is given by\N\[\N[T_\phi]^*_B = \begin{pmatrix} a_0 b_0(0, 0) & a_1 b_1(1, 0) & a_2 b_2(2, 0) & a_3 b_3(3, 0) & \cdots \\\Na_{-1} b_1(1, 0) & \sum_{i=0}^{1} a_{-i} b_1(i, 1) & \sum_{i=1}^{2} a_i b_2(i, 1) & \sum_{i=2}^{3} a_i b_3(i, 1) & \cdots \\\Na_{-2} b_2(2, 0) & \sum_{i=1}^{2} a_{-i} b_2(i, 1) & \sum_{i=0}^{2} a_{-i} b_2(i, 2) & \sum_{i=1}^{3} a_i b_3(i, 2) & \cdots \\\Na_{-3} b_3(3, 0) & \sum_{i=2}^{3} a_{-i} b_3(i, 1) & \sum_{i=1}^{3} a_{-i} b_3(i, 2) & \sum_{i=0}^{3} a_{-i} b_3(i, 3) & \cdots \\\N\vdots & \vdots & \vdots & \vdots & \ddots \\\N\end{pmatrix},\N\]\Nwhere \( b_m(m, n) \in \mathbb{R} \) is denoted as\N\[\N\begin{pmatrix} b_m(m, n) \\\Nb_{m+1}(m, n) \\\Nb_{m+2}(m, n) \\\N\vdots \\\Nb_{m+n}(m, n) \end{pmatrix} = N \begin{pmatrix} N_m(m) N_n(m) \\\NN_m(m+1) N_n(m+1) \\\NN_m(m+2) N_n(m+2) \\\N\vdots \\\NN_m(m+n) N_n(m+n) \end{pmatrix}\N\]\Nfor \( b_j(m, n) \in R\) and \(N\) is denoted as\N\[\NN = \begin{pmatrix} N_m(m) & 0 & 0 & 0 & \cdots & 0 \\\NN_m(m+1) & N_{m+1}(m+1) & 0 & 0 & \cdots & 0 \\\NN_m(m+2) & N_{m+1}(m+2) & N_{m+2}(m+2) & 0 & \cdots & 0 \\\N\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\\NN_m(m+n) & N_{m+1}(m+n) & N_{m+2}(m+n) & N_{m+3}(m+n) & \cdots & N_{m+n}(m+n) \end{pmatrix}.\N\]