A handy formula for the Fredholm index of Toeplitz plus Hankel operators. (English) Zbl 1272.47041
For \(1<p<\infty\), let \(l^p(\mathbb{Z})\) denote the Banach space of all sequences \(x=\{x_n\}_{n\in\mathbb{Z}}\) of complex numbers with the norm \(\|x\|_p=\big(\sum_{n\in\mathbb{Z}}|x_n|^p\big)^{1/p} <\infty\), \(P\) be the canonical projection of \(l^p(\mathbb{Z})\) onto its subspace \(l^p:=l^p(\mathbb{Z}_+)\) of sequences with \(x_n=0\) for \(n<0\), \(Q=I-P\), and let \(J\) act on \(l^p(\mathbb{Z})\) by \((Jx)_n=x_{-n-1}\).
For each function \(a\in L^\infty(\mathbb{T})\), let \(\{a_k\}_{k\in \mathbb{Z}}\) denote the sequence of its Fourier coefficients. The Laurent operator \(L(a)\) associated with \(a\in L^\infty(\mathbb{T})\) acts on the space of all finitely supported sequences on \(\mathbb{Z}\) by \((L(a)x)_k=\sum_{m\in\mathbb{Z}}a_{k-m}x_m\). Let \(M^p\) be the Banach algebra of all \(a\in L^\infty(\mathbb{T})\) for which \(L(a)\) extends to a bounded linear operator on \(l^p(\mathbb{Z})\), and let \(PC_p\) be the Banach subalgebra of \(M^p\) generated by the piecewise constant functions on \(\mathbb{T}\).
Let \(L(X)\) denote the Banach algebra of all bounded linear operators on a Banach space \(X\), and let \(K(X)\) be the ideal of all compact operators in \(L(X)\).
Given \(a\in M^p\), the operators \(T(a):=PL(a)P\) and \(H(a):=PL(a)QJ\) acting on \(l^p\) are called the Toeplitz and Hankel operators with generating function \(a\), respectively. Let \(\text{TH}(PC_p)\) be the Banach subalgebra of \(L(l^p)\) generated by all operators of the form \(T(a)+H(b)\) with \(a,b\in PC_p\).
For \(p\in(1,\infty)\) and \(\lambda\in\overline{\mathbb{R}}\), let \(\mu_p(\lambda):=(1+\coth(\pi\lambda+\pi i/p))/2\) and \(\nu_p(\lambda) :=(2i\sinh(\pi\lambda+\pi i/p))^{-1}\). Put \(\mathbb{T}_+:=\{z\in \mathbb{T}:\text{Im}\,z\geq 0\}\), \(\mathbb{T}_+^0:=\mathbb{T}_+ \setminus\{\pm 1\}\). For \(a\in PC_p\), let \(a(t^+)\) and \(a(t^-)\) denote the right and the left one-sided limits at a point \(t\in\mathbb{T}\).
The Fredholm symbol calculus for the Banach algebra \(\text{TH}(PC_p)\) and the Fredholm criterion for any operator \(A\in\text{TH}(PC_p)\) are given by the following theorem:
(a) Let \(p\in(1,\infty)\), \(a,b\in PC_p\) and \(1/p+1/q=1\). Then the operator \(T(a)+H(b)\) is Fredholm on the space \(l^p\) if and only if the matrix \[ \begin{aligned} &\text{smb}_p(T(a)+H(b))(t,\lambda)\\ &\quad:=\begin{pmatrix} a(t^+)\mu_q(\lambda)+a(t^-)(1-\mu_q(\lambda)) & (b(t^+)-b(t^-))\nu_q(\lambda)\\(b(\bar{t}^-)-b(\bar{t}^+)) \nu_q(\lambda) & a(\bar{t}^-)(1-\mu_q(\lambda))+a(\bar{t}^+) \mu_q(\lambda)\end{pmatrix} \end{aligned} \] is invertible for every \((t,\lambda)\in\mathbb{T}^0_+\times \overline{\mathbb{R}}\) and if the number \[ \begin{aligned} \text{smb}_p(T(a)+H(b))(t,\lambda)&:=a(t^+)\mu_q(\lambda)+a(t^-) (1-\mu_q(\lambda))\\ &\quad+it(b(t^+)-b(t^-))\nu_q(\lambda) \end{aligned} \] is not zero for every \((t,\lambda)\in\{\pm 1\}\times \overline{\mathbb{R}}\).
(b) The mapping \(\text{smb}_p\) defined in assertion (a) extends to a continuous algebra homomorphism from \(\text{TH}(PC_p)\) to the algebra \(\mathcal{F}\) of all bounded functions on \(\mathbb{T}_+ \times\overline{\mathbb{R}}\) with values in \(\mathbb{C}^{2\times 2}\) on \(\mathbb{T}^0_+\times\overline{\mathbb{R}}\) and with values in \(\mathbb{C}\) on \(\{\pm 1\}\times\overline{\mathbb{R}}\). Moreover, there exists a constant \(M\) such that \[ \|\text{smb}_p\,A\|:= \sup_{(t,\lambda)\in\mathbb{T}_+ \times\overline{\mathbb{R}}}\|\text{smb}_p\,A(t,\lambda)\|_\infty\leq M\inf_{K\in K(l^p)}\|A+K\| \] for every operator \(A\in\text{TH}(PC_p)\), where \(\|B\|_\infty\) refers to the spectral norm of the matrix \(B\).
(c) An operator \(A\in\text{TH}(PC_p)\) is Fredholm on the space \(l^p\) if and only if the function \(\text{smb}_p\,A\) is invertible in \(\mathcal{F}\).
(d) The quotient algebra \(\text{TH}(PC_p)/K(l^p)\) is inverse closed in the Calkin algebra \(L(l^p)/K(l^p)\).
With any Fredholm operator \(A\in\text{TH}(PC_p)\), the authors associate the function \(W(A):\mathbb{T}_+\times\overline{\mathbb{R}} \to\mathbb{C}\) defined by \[ W(A)(t,\lambda)=\begin{cases} \text{smb}_p\,A(t,\lambda)/\text{smb}_p\,A (t,\mp\infty) & \text{if }\;t=\pm 1,\\ \det\text{smb}_p\,A(t,\lambda)/(a_{22}(t,+\infty)a_{22}(t,-\infty)) & \text{if }\;t\in\mathbb{T}^0_+,\end{cases} \] where \(\text{smb}_p\,A(t,\lambda)=(a_{ij}(t,\lambda))_{i,j=1}^2\) for \(t\in\mathbb{T}^0_+\). The image of the function \(W(A)\), when \(\lambda\) runs \(\overline{\mathbb{R}}\) from \(-\infty\) to \(+\infty\) at all points \(t_0\in\mathbb{T}_+\) where the function \(W(A)(t_0,\cdot)\) is not constant, and when \(t\) for fixed value \(\lambda=+\infty\) traces all arcs of \(\mathbb{T}_+\) obtained by the partition of \(\mathbb{T}_+\) by the mentioned points \(t_0\), is a closed oriented curve \(\gamma\) in \(\mathbb{C}\) which does not pass through the origin. Hence the winding number \(\text{wind}_{\,\mathbb{T}_+}W(A)\) is well defined as the increment of the argument of the function \(W(A)\) divided by \(2\pi\) when \(W(A)\) traces out the closed oriented curve \(\gamma\).
Finally, it is proved that, if \(A\in\text{TH}(PC_p)\) is a Fredholm operator on the space \(l^p\), then its index can be calculated by the formula \(\text{ind}A=-\text{wind}_{\,\mathbb{T}_+}\,W(A)\).
For each function \(a\in L^\infty(\mathbb{T})\), let \(\{a_k\}_{k\in \mathbb{Z}}\) denote the sequence of its Fourier coefficients. The Laurent operator \(L(a)\) associated with \(a\in L^\infty(\mathbb{T})\) acts on the space of all finitely supported sequences on \(\mathbb{Z}\) by \((L(a)x)_k=\sum_{m\in\mathbb{Z}}a_{k-m}x_m\). Let \(M^p\) be the Banach algebra of all \(a\in L^\infty(\mathbb{T})\) for which \(L(a)\) extends to a bounded linear operator on \(l^p(\mathbb{Z})\), and let \(PC_p\) be the Banach subalgebra of \(M^p\) generated by the piecewise constant functions on \(\mathbb{T}\).
Let \(L(X)\) denote the Banach algebra of all bounded linear operators on a Banach space \(X\), and let \(K(X)\) be the ideal of all compact operators in \(L(X)\).
Given \(a\in M^p\), the operators \(T(a):=PL(a)P\) and \(H(a):=PL(a)QJ\) acting on \(l^p\) are called the Toeplitz and Hankel operators with generating function \(a\), respectively. Let \(\text{TH}(PC_p)\) be the Banach subalgebra of \(L(l^p)\) generated by all operators of the form \(T(a)+H(b)\) with \(a,b\in PC_p\).
For \(p\in(1,\infty)\) and \(\lambda\in\overline{\mathbb{R}}\), let \(\mu_p(\lambda):=(1+\coth(\pi\lambda+\pi i/p))/2\) and \(\nu_p(\lambda) :=(2i\sinh(\pi\lambda+\pi i/p))^{-1}\). Put \(\mathbb{T}_+:=\{z\in \mathbb{T}:\text{Im}\,z\geq 0\}\), \(\mathbb{T}_+^0:=\mathbb{T}_+ \setminus\{\pm 1\}\). For \(a\in PC_p\), let \(a(t^+)\) and \(a(t^-)\) denote the right and the left one-sided limits at a point \(t\in\mathbb{T}\).
The Fredholm symbol calculus for the Banach algebra \(\text{TH}(PC_p)\) and the Fredholm criterion for any operator \(A\in\text{TH}(PC_p)\) are given by the following theorem:
(a) Let \(p\in(1,\infty)\), \(a,b\in PC_p\) and \(1/p+1/q=1\). Then the operator \(T(a)+H(b)\) is Fredholm on the space \(l^p\) if and only if the matrix \[ \begin{aligned} &\text{smb}_p(T(a)+H(b))(t,\lambda)\\ &\quad:=\begin{pmatrix} a(t^+)\mu_q(\lambda)+a(t^-)(1-\mu_q(\lambda)) & (b(t^+)-b(t^-))\nu_q(\lambda)\\(b(\bar{t}^-)-b(\bar{t}^+)) \nu_q(\lambda) & a(\bar{t}^-)(1-\mu_q(\lambda))+a(\bar{t}^+) \mu_q(\lambda)\end{pmatrix} \end{aligned} \] is invertible for every \((t,\lambda)\in\mathbb{T}^0_+\times \overline{\mathbb{R}}\) and if the number \[ \begin{aligned} \text{smb}_p(T(a)+H(b))(t,\lambda)&:=a(t^+)\mu_q(\lambda)+a(t^-) (1-\mu_q(\lambda))\\ &\quad+it(b(t^+)-b(t^-))\nu_q(\lambda) \end{aligned} \] is not zero for every \((t,\lambda)\in\{\pm 1\}\times \overline{\mathbb{R}}\).
(b) The mapping \(\text{smb}_p\) defined in assertion (a) extends to a continuous algebra homomorphism from \(\text{TH}(PC_p)\) to the algebra \(\mathcal{F}\) of all bounded functions on \(\mathbb{T}_+ \times\overline{\mathbb{R}}\) with values in \(\mathbb{C}^{2\times 2}\) on \(\mathbb{T}^0_+\times\overline{\mathbb{R}}\) and with values in \(\mathbb{C}\) on \(\{\pm 1\}\times\overline{\mathbb{R}}\). Moreover, there exists a constant \(M\) such that \[ \|\text{smb}_p\,A\|:= \sup_{(t,\lambda)\in\mathbb{T}_+ \times\overline{\mathbb{R}}}\|\text{smb}_p\,A(t,\lambda)\|_\infty\leq M\inf_{K\in K(l^p)}\|A+K\| \] for every operator \(A\in\text{TH}(PC_p)\), where \(\|B\|_\infty\) refers to the spectral norm of the matrix \(B\).
(c) An operator \(A\in\text{TH}(PC_p)\) is Fredholm on the space \(l^p\) if and only if the function \(\text{smb}_p\,A\) is invertible in \(\mathcal{F}\).
(d) The quotient algebra \(\text{TH}(PC_p)/K(l^p)\) is inverse closed in the Calkin algebra \(L(l^p)/K(l^p)\).
With any Fredholm operator \(A\in\text{TH}(PC_p)\), the authors associate the function \(W(A):\mathbb{T}_+\times\overline{\mathbb{R}} \to\mathbb{C}\) defined by \[ W(A)(t,\lambda)=\begin{cases} \text{smb}_p\,A(t,\lambda)/\text{smb}_p\,A (t,\mp\infty) & \text{if }\;t=\pm 1,\\ \det\text{smb}_p\,A(t,\lambda)/(a_{22}(t,+\infty)a_{22}(t,-\infty)) & \text{if }\;t\in\mathbb{T}^0_+,\end{cases} \] where \(\text{smb}_p\,A(t,\lambda)=(a_{ij}(t,\lambda))_{i,j=1}^2\) for \(t\in\mathbb{T}^0_+\). The image of the function \(W(A)\), when \(\lambda\) runs \(\overline{\mathbb{R}}\) from \(-\infty\) to \(+\infty\) at all points \(t_0\in\mathbb{T}_+\) where the function \(W(A)(t_0,\cdot)\) is not constant, and when \(t\) for fixed value \(\lambda=+\infty\) traces all arcs of \(\mathbb{T}_+\) obtained by the partition of \(\mathbb{T}_+\) by the mentioned points \(t_0\), is a closed oriented curve \(\gamma\) in \(\mathbb{C}\) which does not pass through the origin. Hence the winding number \(\text{wind}_{\,\mathbb{T}_+}W(A)\) is well defined as the increment of the argument of the function \(W(A)\) divided by \(2\pi\) when \(W(A)\) traces out the closed oriented curve \(\gamma\).
Finally, it is proved that, if \(A\in\text{TH}(PC_p)\) is a Fredholm operator on the space \(l^p\), then its index can be calculated by the formula \(\text{ind}A=-\text{wind}_{\,\mathbb{T}_+}\,W(A)\).
Reviewer: Yuri I. Karlovich (Cuernavaca)
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 47L15 | Operator algebras with symbol structure |
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