R. S. ISMAGILOV

ON THE SPECTRUM OF TOEPLITZ MATRICES

(Presented by Academician P. S. Aleksandrov, 23 X 1962)

1. Let \(A\) be a self-adjoint operator in a Hilbert space \(H\), having simple spectrum; \(g \in D_A\) be some generating vector; \(P\) be the operator of orthogonal projection onto the subspace \(Hg=H\ominus g\). In \(Hg\) construct the operator by the formula \(Agx=PAx\) for \(x\in D_A\cap Hg\). At the beginning of the note the spectral properties of \(Ag\) are studied.

If \(A\) has discrete spectrum (we do not consider this case), then a complete description of the spectrum of \(Ag\) is given by the following proposition of M. G. Krein: if \(\lambda_i\) are the eigenvalues of \(A\), \(e_i\) the eigenvectors and \(g=\sum a_i e_i\), then the spectrum of \(Ag\) coincides with the set of roots of the equation

\[ \sum \frac{|a_i|^2}{\lambda_i-z}=0. \]

We begin with the following remark. Let \(\sigma(t)\) be a nondecreasing function on \((-\infty,\infty)\) and \(\int t^2\,d\sigma(t)<\infty\). Put

\[ f(z)=\int \frac{d\sigma(t)}{t-z}\qquad (\operatorname{Im} z>0). \tag{1} \]

Obviously, \(\operatorname{Im}\left(-\frac{1}{f(z)}\right)\geq 0\) for \(\operatorname{Im} z>0\); using the well-known theorem of Herglotz ((\(^{1}\)), p. 117) and the condition \(\int t^2\,d\sigma(t)<\infty\), it is easy to show that

\[ -\frac{1}{f(z)}=\gamma z+\beta+\int \frac{d\sigma^*(t)}{t-z}, \tag{2} \]

where \(\sigma^*(t)\) is a nondecreasing function, uniquely determined by the function \(\sigma(t)\) by formulas (1) and (2). We now pass to the description of the spectrum of \(Ag\).

Theorem 1. Let \(E_t\) be the spectral family of the operator \(A\), \(\sigma(t)=(E_tg,g)\), and let \(\sigma^*(t)\) be determined from \(\sigma(t)\) by formulas (1) and (2). Then \(Ag\) is unitarily equivalent to the operator of multiplication by the independent variable in the space \(L_2(\sigma^*,-\infty,\infty)\).

As is known, every spectral family \(E_t\) in \(H\) is uniquely decomposed into a sum \(E_t^a+E_t^c\), where \(E_t^a\) is a weakly absolutely continuous function, and \(E_t^c\) is a weakly singular* function of \(t_0\).

Theorem 2. Let \(E_t\) and \(F_t\) be the spectral families of the operators \(A\) and \(Ag\); \(E_t^a\) and \(F_t^a\) their absolutely continuous, and \(E_t^c\) and \(F_t^c\) their singular components. Then \(E_t^a\) and \(F_t^a\) are unitarily equivalent, and \(E_t^c\) and \(F_t^c\) are mutually singular.

Let us explain that the mutual singularity of \(E_t^c\) and \(F_t^c\) means the mutual singularity of the scalar measures corresponding to the functions \((E_t^c h,h)\) and \((F_t^c g,g)\) for any \(h\in H\), \(g\in Hg\).

We outline the proofs of the theorems.

* This means that for any \(h\in H\) the measure \(d(E_t^a h,h)\) is absolutely continuous, while \(d(E_t^c h,h)\) is singular; here the singular function is not necessarily discontinuous.

One may assume that \(H=L_2(\sigma)\), \(g=g(t)\equiv 1\), \(Af=tf(t)\). Let first \(\|A\|<\infty\), and hence \(d\sigma\) is finite. In the basis consisting of the orthogonal polynomials \(P_n(t)\) \((n\geqslant 0)\) with respect to \(d\sigma\), the operator \(A\) is represented by the Jacobi matrix \(J\), and the operator \(Ag\) by the matrix \(J_1\), obtained from \(J\) by deleting the first row and the first column. From the matrix \(J_1\) we construct in the known way the measure \(d\sigma^*\) and the orthogonal polynomials \(P_n^*(t)\) \((n\geqslant 0)\). Then the correspondence \(P_n(t)\leftrightarrow P_{n+1}^*(t)\) gives rise to an isometric correspondence \(U\) between \(L_2(\sigma)\ominus g\) and \(L_2(\sigma^*)\), under which \(Ag\) goes over into multiplication by \(t\) in \(L_2(\sigma^*)\). Constructing the functions \(f(z)=\int \dfrac{d\sigma(t)}{t-z}\) and \(f^*(z)=\int \dfrac{d\sigma^*(t)}{t-z}\), and, by the known formula for the expansion of these functions into continued fractions ((1), p. 34), finding
\[ f^*(z)=\gamma z+\beta-\frac{1}{f(z)}, \]
we obtain formulas (1) and (2), and this proves the theorem for the case \(\|A\|<\infty\).

From the properties of orthogonal polynomials it is easy to obtain that the mapping \(U\) introduced above is given by the formula
\[ Uf=\int \frac{f(x)-f(t)}{x-t}\,d\sigma(t) \tag{3} \]
and that
\[ U^{-1}\varphi=\int \frac{\varphi(x)-\varphi(t)}{x-t}\,d\sigma^*(t)+(x-a_0)\varphi(t). \tag{4} \]

To prove the theorem in the case \(\|A\|=\infty\), we first prove, by passage to the limit from finite measures, that formulas (3) and (4) establish an isometric correspondence between \(L_2(\sigma)\ominus g\) and \(L_2(\sigma^*)\); then we prove that \(U\) takes \(Ag\) into multiplication by \(t\) in \(L_2(\sigma^*)\).

To prove Theorem 2, put \(\operatorname{Im} z\to 0\) in (1) and (2). Then
\[ \frac{d\sigma(x)}{dx}=|f(x+i0)|^2\frac{d\sigma^*(x)}{dx}. \]

Therefore \(0<\sigma'(x)/\sigma^{*'}(x)<\infty\) almost everywhere, which is equivalent to the first assertion of Theorem 2.*

From the easily proved inequalities**
\[ \operatorname{Im} f(z)>\frac{\sigma(x+y)-\sigma(x-y)}{2y},\qquad \operatorname{Im} f^*(z)>\frac{\sigma^*(x+y)-\sigma^*(x-y)}{2y} \]
and (1), (2), we obtain
\[ \frac{\sigma(x+y)-\sigma(x-y)}{2y}\, \frac{\sigma^*(x+y)-\sigma^*(x-y)}{2y}<1. \]

From this inequality one easily obtains the second assertion of Theorem 2.

II. Let us apply Theorem 2 to Toeplitz matrices. If \(F(x)\ne \mathrm{const}\), \(F(x)\in L_2(-\pi,\pi)\) and \(F(x)=\sum c_n e^{inx}\), then the matrix \(\{a_{ij}\}=\{c_{i-j}\}\) \((i,j\geqslant 0)\) is called a Toeplitz matrix. In the space \(l_2\) of one-sided sequences \(x=\{x_k\}\) \((k\geqslant 0)\), define the operator \(T_F^0\), setting, for finite
\[ x=\{x_k\}\quad T_F^0x=y,\quad \text{where } y=\{y_n\},\quad y_n=\sum_0^\infty c_{n-j}x_j\quad (n\geqslant 0). \]
Let \(T_F\) be the closure of the operator \(T_F^0\). The spectral properties of the operator \(T_F\) were studied in \((^3,^5,^8)\).

* This assertion also follows from Kato’s theorem on finite-dimensional perturbations \((^4)\).
** A similar inequality is found in Fatou \((^3)\).

The principal known results are as follows:

1) If \(F(x)\) is semibounded, then \(T_F\) is self-adjoint \((^{2})\).

2) If \(T_F\) is self-adjoint, then its spectrum is continuous and fills the interval \((\inf F, \sup F)\) \((^{2})\).

3) If \(F(x)\) is semibounded, then \(T_F\) has an absolutely continuous spectrum \((^{8})\).

4) If \(F(x)\) is an even periodic function, \(F'(x)\) exists and is expandable in an absolutely convergent Fourier series, then \(T_F\) is unitarily equivalent to the absolutely continuous component of the operator of multiplication by \(F(x)\) in \(L_2(0,\pi)\) \((^{8})\).

Below we give a complete spectral description of the operator \(T_F\) for arbitrary \(F(x)\in L_2(-\pi,\pi)\).

Theorem 3. The operator \(T_F\) has an absolutely continuous spectrum if and only if it is self-adjoint.

To clarify the idea of the proof, let us consider the simplest case: suppose \(T_F\) has simple spectrum and the vector \(e_0=(1,0,0,\ldots)\) is a generating vector. Let \(\widehat T_F\) be defined in \(H=l_2\ominus e_0\) by the formula \(\widehat T_F x=PT_Fx\), where \(P\) is the projection onto \(H\). Obviously, \(T_F\) and \(\widehat T_F\) are unitarily equivalent; but, by Theorem 2, their singular components are mutually singular. Thus \(T_F\) cannot have a singular component. In the general case the proof also uses Theorem 2, but requires rather cumbersome geometric considerations.

For what follows we shall need the following.

Definition 1. Let \(N_F(\lambda)\) be a function of \(\lambda\) defined as follows: if the set \(E\{x:F(x)<\lambda\}\) consists of a finite number of intervals mod 0, then \(N_F(\lambda)\) is equal to the number of these intervals; otherwise \(N_F(\lambda)=\infty\).

In defining \(N_F(\lambda)\) one should take into account that \(F(x)\) is considered on the interval \((-\pi,\pi)\) with the endpoints \(-\pi\) and \(\pi\) identified. Note that for smooth functions \(F(x)\), the function \(2N_F(\lambda)\) coincides with the Banach indicatrix of the function \(F(x)\). The following theorem gives a complete spectral description of the operator \(T_F\).

Theorem 4. Let \(E_m\) be the set on the \(\lambda\)-axis where \(N_F(\lambda)=m\) \((m=1,2,\ldots,\infty)\), and let \(A_m\) be the operator of multiplication by an independent variable in \(L_2(E_m)\). Further, let \(B_m=A_m\oplus\cdots\oplus A_m\) (the sum contains \(m\) copies of the operator \(A_m\)) and \(A_F=B_1\oplus\cdots\oplus B_\infty\). Then \(T_F\) is unitarily equivalent to the operator \(A_F\).

Let us consider the simplest case: \(N_F(\lambda)=1\) for \(\inf F<\lambda<\sup F\).

Map \(l_2(0,\infty)\) onto the space \(H_2\) of functions \(f(z)\),

\[ f(z)=\sum_0^\infty a_n z^n \]

\((|z|<1)\), with metric

\[ \|f\|^2=\sum_0^\infty |a_n|^2. \]

Let \(e_k=z^k\) and \(R_\lambda=(T_F-\lambda E)^{-1}\). One can find

\[ \chi_k(\sigma,z)=\lim_{\varepsilon\to 0}\bigl(R_{\sigma+i\varepsilon}e_k-R_{\sigma-i\varepsilon}e_k\bigr). \]

It turns out that \(\chi_k(\sigma,z)=p_k(\sigma)\chi_0(\sigma,z)\), and \(\chi_k(\sigma,z)\ne 0\) for almost all \(\sigma\). Hence it follows that \(T_F\) has a simple Lebesgue spectrum. An analogous consideration is used for the proof in the general case.

For some functions \(F(x)\), the description of the spectrum of the operator \(T_F\) can be carried out without Theorem 4; for this purpose the following theorem is used:

Theorem 5. If \(A\), \(B\), and \(C=A+B\) are self-adjoint bounded operators in \(H\); \(A^a\), \(B^a\), \(C^a\) are their absolutely continuous components, and the operator \(AB\) is nuclear, then \(C^a\) is unitarily equivalent to \(A^a\oplus B^a\).

It is easy to show that if \(\Phi_1(x)\) and \(\Phi_2(x)\) are functions concentrated on disjoint closed sets, then \(T_{\Phi_1}\cdot T_{\Phi_2}\) is a nuclear operator and, by Theorem 5, \(T_{\Phi_1+\Phi_2}\) is equivalent to \(T_{\Phi_1}\oplus T_{\Phi_2}\). Repeated application of this remark to the operator \(T_F\) makes it possible in a number of cases to reduce the problem of the spectrum to the simplest case, when the operator has a simple spectrum (see above).

Theorem 6. The operator \(T_F\) is self-adjoint if, in some neighborhood of each point \(x\in(-\pi,\pi)\) of the interval (with the endpoints \(-\pi\) and \(\pi\) identified), the function \(F(x)\) is semibounded.

Finally, let us note that for two extensions \(A_1\) and \(A_2\) of a simple symmetric operator \(A\), a theorem analogous to Theorem 2 can be proved: the singular components \(A_1\) and \(A_2\) are mutually singular, while the absolutely continuous ones are equivalent. From this one can obtain Putnam’s results \((^6,^7)\).

Note added in proof. After the present paper had been submitted for publication, I learned that the result concerning the spectra of two extensions of a symmetric operator had been obtained earlier by Aronszajn \((^9)\), although only as applied to the Sturm–Liouville operator.

Moscow State University
named after M. V. Lomonosov

Received
19 X 1962

REFERENCES

\(^1\) N. I. Akhiezer, The Classical Moment Problem, Moscow, 1961, p. 35.
\(^2\) P. Hartman, A. Wintner, Am. J. Math., 72, 359 (1950).
\(^3\) P. Fatou, Acta Mat., 30, 335 (1906).
\(^4\) P. Kato, J. Math. Soc. Japan, 4, No. 3 (1952).
\(^5\) C. R. Putnam, Trans. Am. Math. Soc., 87, 2 (1958).
\(^6\) C. R. Putnam, Pacif. J. Math., 9, 3 (1950).
\(^7\) C. R. Putnam, Canad. J. Math., 6, 420 (1954).
\(^8\) M. Rosenblum, Pacif. J. Math., 10, 3 (1958).
\(^9\) N. Aronszajn, Am. J. Math., 79, No. 3, 611 (1957).