UDC 517.948+513.88

MATHEMATICS

A. L. CHISTYAKOV

INFINITE HERMITIAN MATRICES WITH OPERATOR COEFFICIENTS

(Presented by Academician I. G. Petrovskii, 11 IV 1967)

This article is devoted to the study of the deficiency indices of symmetric operators generated in a certain Hilbert space \(\mathfrak H\) by infinite Hermitian matrices with operator coefficients.

1. Let \(\{H_k\}_{k=0}^{\infty}\) be a sequence of Hilbert spaces with scalar products \((\cdot,\cdot)_k\) and norms \(\|\cdot\|_k\). The Hilbert space \(\mathfrak H\) is defined as the direct sum

\[ \mathfrak H=\sum_{k=0}^{\infty}\oplus H_k. \]

The elements of this space are written in the form of sequences \(U=\{u_0,u_1,\ldots,u_k,\ldots\}\), where \(u_k\in H_k\). The scalar product in \(\mathfrak H\) is defined by the formula

\[ [U,V]=\sum_{k=0}^{\infty}(u_k,v_k)_k. \]

Linear operators in \(\mathfrak H\) are specified by infinite matrices

\[ C=(C_{jk})_{j,k=0}^{\infty} \]

with operator coefficients, where \(C_{jk}\) is a linear operator with domain of definition \(D(C_{jk})\), dense in the space \(H_k\), and range in \(H_j\). Moreover, for all \(j\) and \(k\), \(D(C_{jk})\supset D(C_{kk})\). The matrices are assumed to be Hermitian and finite. The Hermitian property of \(C\) means that \(C_{kj}=C_{jk}^{*}\) (where \(C_{jk}^{*}\) is the operator adjoint to \(C_{jk}\)). Finiteness means that for each \(j\) there exists an integer \(m_j\ge 0\) such that \(C_{jk}=0\) for all \(k\) satisfying \(|j-k|>m_j\). Matrices for which \(m_j\equiv m<\infty\) and for all \(j\) there exist bounded operators \(C_{j+m,j}^{-1}\) will be called operator \(C_m\)-matrices.

Defined on the set of finite vectors* with components \(u_k\in D(C_{kk})\), the correspondence \(U\to CU\) generates a symmetric operator in \(\mathfrak H\). Its closure in \(\mathfrak H\) will be denoted by \(C\). The operator \(C\), obviously, is symmetric with a domain of definition dense in \(\mathfrak H\).

Below we study the deficiency indices of the operator \(C\) depending on the properties of the operators \(C_{jk}\). The formulated theorems find application in the consideration of operators in the space of secondary quantization.

2. Known sufficient conditions for self-adjointness of the operator \(C\) ((\(^{1}\)), p. 151; (\(^{2}\))) are a generalization of T. Carleman’s condition, formulated for ordinary Jacobi matrices (\(C_1\)-matrices with numerical coefficients). The theorem given below is a supplement to the results of (\(^{1,2}\)) in the case when the operators \(C_{k,k+j}\) \((j\ne 0)\) are unbounded for every \(j\ne 0\), or are bounded and for at least one \(j\ne 0\) satisfy the condition

\[ \sum_{k=0}^{\infty}\|C_{k,k+j}\|^{-1}<\infty. \]

* A vector is called finite if only a finite number of its components are different from zero.

Let \(I_k\) be the identity operator in \(H_k\), and \(\delta_{jk}\) the Kronecker symbol.

Theorem 1. The operator \(C\) is self-adjoint if there exists a complex number \(\sigma\) such that the quantities
\[ \mu_{jk}=\|C_{jk}(C_{kk}-\sigma I_k)^{-1}\| \]
satisfy the estimates
\[ \sup_k \sum_{j=0}^{\infty}(1-\delta_{jk})\mu_{jk}<1,\qquad \sup_j \sum_{k=0}^{\infty}(1-\delta_{jk})\mu_{jk}<1 \]
and, uniformly in \(k\),
\[ \sum_{j=k+N}^{\infty}\mu_{jk}\to 0 \quad \text{as } N\to\infty . \]

The proof of Theorem 1 is based on the well-known lemma of T. Kato \((^4)\). It asserts that, under a perturbation of a self-adjoint operator \(T_0\) by a symmetric operator \(V\), the property of self-adjointness is preserved if the domains of definition of the operators \(T_0\) and \(V\) are related by the inclusion \(D(T_0)\subset D(V)\), and there exist constants \(a\) and \(b\) \((0\le a<1,\ b>0)\) such that for every \(f\in D(T_0)\) the inequality
\[ \|Vf\|\le a\|T_0 f\|+b\|f\| \]
holds.

An illustration of Theorem 1 may be the case when \(H_k=L_2(-\infty,\infty)\) for all \(k\ge 0\) and the operator \(C\) is generated by an operator \(C_m\)-matrix: \(C_{kk}=-d^2/dx^2\), while for \(k<j\le k+m\)
\[ C_{kj}+\varphi_{kj}(x)\frac{d}{dx}+\psi_{kj}(x) \]
and \(C_{jk}=C_{kj}^*\). All the indicated operators are defined in \(L_2(-\infty,\infty)\) on closures of finite functions, and the complex-valued functions \(\varphi_{kj}(x)\), \(\varphi'_{kj}(x)\), and \(\psi_{kj}(x)\) belong to \(L_2(-\infty,\infty)\), and their norms are uniformly bounded with respect to all \(k\) and \(j\).

1. The presence of nonzero deficiency indices of the operator \(C\) is established for the case when all spaces \(H_k\) coincide and all operators \(C_{jk}\) are bounded. Put \(\dim H_k=d\le\infty\). The following theorem generalizes the corresponding results of papers \((^5,^6)\).

Theorem 2. Let the elements of the \(C_m\)-matrix be such that:
\[ 1)\quad \sum_{k=0}^{\infty}\|C_{k,k+m}^{-1}\|<\infty; \]
\[ 2)\quad \text{for sufficiently large } k \qquad \|C_{k,k+m}^{-1}C_{k,k-m}\|\le 1-\nu/k,\qquad \text{where } \nu>m; \]
\[ 3)\quad \sum_{k=0}^{\infty}\|C_{k,k+m}^{-1}C_{k,k+j}\|<\infty \quad \text{for } |j|<m. \]

Then the operator \(C\) has deficiency indices \((dm,dm)\).

1. Let now \(C\) belong to the class of \(C_1\)-matrices with matrix coefficients (\(m\times m\)-matrices), i.e. have the form
   \[ C_0= \begin{pmatrix} A_0 & B_2 & & \\ B_0^* & A_1 & B_1 & \\ & & \ddots & \ddots \\ & & B_1^* & \ddots \\ & & & \ddots \end{pmatrix}, \tag{1} \]
   where the unspecified elements are zero, and \(\det B_k\ne 0\) \((k=0,1,\ldots)\). For such matrices, called by M. G. Krein \((^7)\) \(J_m\)-matrices, it is possible to supplement the results formulated in Theorems 1 and 2.

Let us introduce into consideration the equation with respect to \(w(k)\)
\[ \det\,[B_{k-1}^*+(A_k-\lambda I)w(k)+B_k w^2(k)]=0, \tag{2} \]
where \(\lambda\) is a complex number, and \(I\) is the identity matrix of order \(m\). Number the roots of equation (2) in the order of increase of their moduli:
\[ |w_1(k)|\le |w_2(k)|\le \cdots \le |w_{2m}(k)|. \]
Denote by \(n_+\) (\(n_-\)) the number of solutions \(w(k)\) of equation (2) for \(\lambda=i\) (\(\lambda=-i\)) satisfying, for all sufficiently large \(k\), the estimate \(|w(k)|\le 1-\nu/k\), and by \(p_+\) (\(p_-\)) —

the number of solutions with the estimate \(|w(k)| \geq 1-1/2k\). In view of the condition \(\det B_k \neq 0\), equation (2) is equivalent to the equation

\[ \det \left[ w^2(k) I+w(k)F_1(k;\lambda)+F_2(k)\right]=0, \]

where \(F_1(k;\lambda)=B_k^{-1}(A_k-\overline{\lambda}I)\), \(F_2(k)=B_k^{-1}B_{k-1}^{*}\).

Theorem 3. Suppose: 1) the elements of the matrices \(F_1(k+1;\pm i)-F_1(k;\pm i)\) and \(F_2(k+1)-F_2(k)\) are absolutely summable on the interval \((0 \leq k \leq \infty)\); 2) the limiting values of all \(2m\) roots \(w_j=\lim_{k\to\infty} w_j(k)\) are nonzero and distinct; 3) there exists \(\nu>1/2\) such that \(n_\pm^\nu+p_\pm=2m\). Then the defect indices of the operator \(C\) are equal to \((m-p_+,m-p_-)\).

The proof of Theorem 3 is based on computing asymptotic estimates, as \(k\to\infty\), for the components of the vector \(u_k=(x_{km},x_{km+1},\ldots,x_{km+m-1})\), which is a solution of the equation

\[ B_{k-1}^{*}u_{k-1}+A_ku_k+B_ku_{k+1}=\overline{\lambda}u_k \quad (k \geq 1). \tag{3} \]

The plan of the proof is as follows: replace the system of second-order difference equations (3) by a system of first-order difference equations \(y(k+1)=G_ky(k)\) (where \(y(k)\) is a \(2m\)-dimensional vector), and reduce the resulting system to quasidiagonal form

\[ t(k+1)=(\Gamma_k+\Gamma_kD_k)t(k), \tag{4} \]

where \(\Gamma_k\) is a diagonal matrix, and the elements \(d_{ij}(k)\) of the matrix \(D_k\) satisfy the condition

\[ \sum_{k=0}^{\infty} |d_{ij}(k)|<\infty. \]

After this, it remains to use I. M. Rapoport’s theorem (\((^8)\), p. 62) on the asymptotics of the solutions of equation (4) and to compute, for large \(k\), the asymptotics of the products \(\prod_{s=0}^{k} w_j(s)\), where \(j=1,2,\ldots,2m\). This makes it possible to determine the number of solutions of equation (3) belonging to \(l_2\).

Let us now consider the operator \(C_0\), defined as the restriction of the operator \(C\) by means of the additional condition \(u_0=0\). The defect subspace \(T_\lambda^0\) of the operator \(C_0\) consists of the quadratically summable solutions of equation (3). The dimension of the defect subspace \(T_\lambda\) of the operator \(C\) is determined by the formula \(\dim T_\lambda^0=\dim T_\lambda+m\), obtained on the basis of a lemma of I. M. Glazman (\((^9)\), p. 47).

A particular case of \(J_m\)-matrices is given by \(C_m\)-matrices with numerical coefficients (see Sec. 1). However, instead of system (3), for \(C_m\)-matrices it is more convenient to write a single difference equation of order \(2m\)

\[ \sum_{j=k-m}^{k+m} c_{kj}x_j=\overline{\lambda}x_k. \]

The corresponding characteristic equation has the form

\[ \sum_{s=0}^{2m} f_s(k) w^{2m-s}(k)=0, \tag{5} \]

where \(f_s(k)=c_{k,k+m-s}/c_{k,k+m}\) for \(s\neq m\), and \(f_m^{(k)}=(c_{kk}-\overline{\lambda})/c_{k,k+m}\). We number the roots of equation (5) in increasing order of their moduli and introduce the numbers \(n_\pm^\nu\) and \(p_\pm\) exactly as was indicated above.

Theorem 4. Suppose the elements of the \(C_m\)-matrix are such that:

1) \[ \sum_{k=0}^{\infty} |f_s(k+1)-f_s(k)|<\infty \quad \text{for } s=1,2,\ldots,2m; \]

2) the limiting values of all \(2m\) roots \(w_j=\lim_{k\to\infty} w_j(k)\) of equation (5) are nonzero and distinct;

3) there exists \(\nu > 1/2\) such that \(n_{\pm}^{\nu}+p_{\pm}=2m\).

Then the deficiency indices of the operator \(\widetilde C\) are \((m-p_{+}, m-p_{-})\).

As examples, let us consider operators \(C\) generated by \(C_m\)-matrices with elements of the form \(c_{k,k+j}=a_j k^{\alpha_j}\) \((j=0,1,\ldots,m)\). If \(\alpha_0=\alpha_m>\alpha_i+1\) \((i=1,2,\ldots,m-1)\) and \(2|a_m|>|a_0|\), then \(C\) has deficiency indices \((m,m)\), although it does not satisfy the conditions of Theorem 2. As an example of an operator with deficiency indices \((\rho,\rho)\), where \(0<\rho<m\), one may take the case: \(m=2\), \(c_{kk}=(a+2a/k)k^\alpha\), \(c_{k,k+1}=bk^\alpha\), \(c_{k,k+2}=k^\alpha\), \(\alpha>1\), \(|a+2|<2|b|\), \(a<b^2/4+2\). The corresponding operator \(C\) has deficiency indices \((1,1)\).

For ordinary Jacobi matrices the following theorem holds.

Theorem 5. Let \(C\) be the operator generated by a Jacobi matrix with elements \(a_k=ak^\alpha\) and \(b_k=bk^\beta\) (\(a\) and \(\alpha\) are real, \(\beta>1\), \(b\) is complex). Then: 1) \(C\) is self-adjoint if \(\alpha>\beta\), or if \(\alpha=\beta\) and \(|a|\ge 2|b|\); 2) \(C\) has deficiency indices \((1,1)\) if \(\alpha<\beta\), or if \(\alpha=\beta\) and \(|a|<2|b|\).

5. The results presented find application in clarifying the question of self-adjointness of certain operators in the space of second quantization. Let, as usual, \(a^*(p)\) and \(a(p)\) be the creation and annihilation operators satisfying the Bose or Fermi commutation relations. The spaces \(H_k\) are symmetrized or antisymmetrized tensor products of \(k\) copies of the space \(L_2(E_3)\), where \(E_3\) is three-dimensional Euclidean space. With the aid of Theorem 1, for example, one verifies the self-adjointness of the operator \(T=T_0+V+V^*\), where

\[ T_0=\int_{E_3}\omega(p)a^*(p)a(p)\,dp +\int_{E_3}\int_{E_3}\varphi(p_1,p_2)a^*(p_1)a^*(p_2)a(p_1)a(p_2)\,dp_1dp_2, \]

\[ V=\int_{E_3}\int_{E_3}\int_{E_3}\int_{E_3} v(p_1,p_2,p_3\mid q)a^*(p_1)a^*(p_2)a^*(p_3)a(q)\,d^3p\,dq, \]

\[ \omega(p)\ge 0,\qquad \varphi(p_1,p_2)\ge \delta>0,\qquad v\in L_2(E_{12}),\qquad \|v\|\delta^{-1}<1/2. \]

When considering the model of a system with one degree of freedom, i.e., when the argument \(p\) assumes a single value, one can use Theorem 4. For example, the operator \(C\) generated by the expression

\[ a^*a+\varepsilon(\overline{\gamma}_2 a^{*4}+\overline{\gamma}_1 a^{*3}a+\gamma_0 a^{*2}a^2+\gamma_1 a^*a^3+\gamma_2 a^4) \]

is self-adjoint for arbitrary real \(\varepsilon\) and \(\gamma_0\), if the roots of the equation
\[ \gamma_2 w^4+\gamma_1 w^3+\gamma_0 w^2+\gamma_1 w+\overline{\gamma}_2=0 \]
are distinct and different from zero, and the moduli of two of them are less than 1 and of two are greater than 1. This holds if \(|\gamma_0|\ge 2|\gamma_1|+2|\gamma_2|\). The deficiency indices of such an operator will be \((2,2)\), if \(\gamma_1=0\) and \(|\gamma_0|<2|\gamma_2|\).

Taking this opportunity, I express my deep gratitude to F. A. Berezin and B. M. Levitan.

Institute of Organoelement Compounds
Academy of Sciences of the USSR

Received
10 III 1967

REFERENCES

1. F. A. Berezin, The Method of Second Quantization, “Nauka,” 1965.
2. V. G. Tarnopol’skii, Dokl. Akad. Nauk USSR, No. 11, 1189 (1959).
3. N. I. Akhiezer, The Classical Moment Problem, Moscow, 1961.
4. T. Kato, Trans. Am. Math. Soc., 70, 195 (1951).
5. Yu. M. Berezanskii, Tr. Mosk. Mat. Obshch., 5, 203 (1956).
6. V. G. Tarnopol’skii, Dokl. Akad. Nauk USSR, No. 3, 305 (1960).
7. M. G. Krein, DAN, 69, No. 2, 125 (1949).
8. I. M. Rapoport, On Certain Asymptotic Methods in the Theory of Differential Equations, Kiev, 1954.
9. I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Moscow, 1963.