Abstract Generated abstract
This paper studies multiplicative representations of analytic operator functions in a separable Hilbert space that are holomorphic off a bounded real set, have a compact-operator expansion at infinity, and satisfy a J-nonexpansive positivity condition in the upper half-plane. Extending earlier finite-trace results of Livshits and Ginzburg, it proves such a representation under a weaker summability condition on the eigenvalues of an associated compact self-adjoint operator, using methods from the theory of non-self-adjoint operators and triangular representations. A separate theorem treats functions entire in 1/lambda, where the summability condition is unnecessary. The resulting integral product formulas are shown to converge in norm and to satisfy related integral equations, semigroup relations, and positivity identities.
Full Text
M. S. BRODSKII
ON THE MULTIPLICATIVE REPRESENTATION OF SOME ANALYTIC OPERATOR FUNCTIONS
(Presented by Academician V. I. Smirnov on 9 I 1961)
- In the works of V. P. Potapov \((^{1,2})\) the problem of the multiplicative representation of analytic and \(J\)-nonexpanding matrix functions in the upper half-plane was solved. Subsequently, M. S. Livshits \((^3)\) and Yu. P. Ginzburg \((^4)\) investigated certain generalizations of this problem to the infinite-dimensional case.
Consider an operator function \(W(\lambda)\) possessing the following properties:
I. The function \(W(\lambda)\) is holomorphic in the domain \(G\), obtained by removing from the extended complex plane a certain bounded set of real points.
II. In a neighborhood of the infinitely distant point the function \(W(\lambda)\) is expanded in a norm-convergent series
\[ W(\lambda)=E+\frac{1}{\lambda}W_1+\frac{1}{\lambda^2}W_2+\cdots, \]
where \(W_k\) \((k=1,2,\ldots)\) are completely continuous operators acting in the separable Hilbert space \(\mathfrak H_W\).
III. There exists an operator \(J\) such that \(J=J^*\); \(J^2=E\); \(W^*(\lambda)JW(\lambda)-J\geqslant 0,\ \operatorname{Im}\lambda>0\); \(W^*(\lambda)JW(\lambda)-J=0,\ \operatorname{Im}\lambda=0,\ \lambda\in G\).
Let us note that from II and III it follows that the operator \(T\), defined by the equality \(W_1=2iTJ\), is positive.
The theorems of M. S. Livshits and Yu. P. Ginzburg make it possible to represent the operator function \(W(\lambda)\) multiplicatively only in the case when the trace of the operator \(T\) converges. More general assertions, formulated below, are obtained by means of methods from the theory of non-self-adjoint operators.
Theorem 1. If the operator function \(W(\lambda)\) possesses properties I, II, III and satisfies the condition *
\[ \sum_n \frac{|\omega_n|}{n}<\infty, \tag{1} \]
where \(\omega_n\) \((n=1,2,\ldots)\) are the nonzero eigenvalues of the operator \(T^{1/2}JT^{1/2}\), numbered with multiplicities taken into account in decreasing order of absolute values, then
\[ W(\lambda)=\int_0^1 \left(E+\frac{2i}{\lambda-\varphi(x)}\,dF(x)J\right), \tag{2} \]
\[ \text{*} \]
The class of completely continuous self-adjoint operators satisfying condition (1) was investigated by V. I. Matsaev. The proof of Theorem 1 is based on the author’s work \((^6)\), in which the results of V. I. Matsaev found an essential application.
where \(\varphi(x)\) is a left-continuous nondecreasing scalar function, and \(F(x)\) is a strictly increasing absolutely continuous operator-function with completely continuous values, normalized by the condition \(F(0)=0\). The integral products
\[ \left(E+\frac{2i}{\lambda-\varphi(\xi_1)}\Delta F_1J\right) \left(E+\frac{2i}{\lambda-\varphi(\xi_2)}\Delta F_2J\right)\cdots \]
\[ \cdots \left(E+\frac{2i}{\lambda-\varphi(\xi_n)}\Delta F_nJ\right) \tag{3} \]
\[ (0=x_0<\xi_1\le x_1<\xi_2\le x_2<\cdots<\xi_n\le x_n=1,\qquad \Delta F_k=F(x_k)-F(x_{k-1})) \]
converge in norm to \(W(\lambda)\) in the sense of C. O. Shatunovskii.
Theorem 2. If \(W(\lambda)\) is an entire function of \(1/\lambda\) possessing properties II and III, then
\[ W(\lambda)=\int_0^1\left(E+\frac{2i}{\lambda}\,dF(x)J\right), \tag{4} \]
where \(F(x)\) is a strictly increasing absolutely continuous operator-function with completely continuous values, normalized by the condition \(F(0)=0\). The integral products
\[ \left(E+\frac{2i}{\lambda}\Delta F_1J\right) \left(E+\frac{2i}{\lambda}\Delta F_2J\right)\cdots \left(E+\frac{2i}{\lambda}\Delta F_nJ\right) \tag{5} \]
\[ (0=x_0\le \xi_1\le x_1\le \xi_2\le x_2\le\cdots\le \xi_n\le x_n=1,\qquad \Delta F_k=F(x_k)-F(x_{k-1}) \]
converge in norm to \(W(\lambda)\) as \(\max (x_k-x_{k-1})\to 0\).
- Let us briefly dwell on the considerations which lead to the proof of Theorem 1. Generalizing the method of M. S. Livshits \((^3)\), we introduce the function
\[ V(\lambda)=i(W(\lambda)+E)^{-1}(W(\lambda)-E)J. \]
It is easy to verify that it is holomorphic in the upper half-plane and has nonnegative imaginary part
\[ \frac{V(\lambda)-V^*(\lambda)}{2i}, \]
equal to zero at real points lying outside some interval \((a,b)\). Since, moreover, \(\lambda V(\lambda)\) tends to a bounded operator as \(\lambda\to\infty\), there exists a nondecreasing function \(\sigma(x)\) with completely continuous values such that \((^5)\)
\[ V(\lambda)=\int_a^b \frac{d\sigma(t)}{t-\lambda} \qquad (\operatorname{Im}\lambda\ne 0,\ \sigma(0)=0). \]
Modifying somewhat the proof of a known theorem of M. A. Naimark, we represent the function \(\sigma(t)\) in the form \(\sigma(t)=R^*\mathcal E(t)R\), where \(R\) is a completely continuous mapping of the space \(\mathfrak H\) into some Hilbert space \(\mathfrak H_0\), and \(\mathcal E(t)\) is an orthogonal resolution of the identity in \(\mathfrak H_0\). Consider in \(\mathfrak H_0\) the operator
\[ A_0=\int_a^b t\,d\mathcal E(t)+iRJR^*. \tag{6} \]
Analysis of formula (6) shows that the spectrum of the operator \(A_0\) is the set of singular points of the function \(W(\lambda)\). Direct computation of the resolvent of the operator \(A_0\) leads to the formula
\[ W(\lambda)=E-2iR^*(A_0-\lambda E)^{-1}RJ. \tag{7} \]
In an arbitrary Hilbert space \(\mathfrak H_1\) define a Hermitian operator \(A_1\) so that its spectrum is a part of the spectrum of the operator \(A_0\),
and introduce the operator \(A=A_0\oplus A_1\). We then extend the domain of definition of the operator \(R^*\) to the whole space \(\mathfrak H=\mathfrak H_0\oplus\mathfrak H_1\), assuming that \(R\) maps \(\mathfrak H_W\) into \(\mathfrak H\).
It is obvious that the operator \(A\) has a completely continuous imaginary part
\[ \frac{A-A^*}{2i}=RJR^* \]
and a purely real spectrum. Since \(T=R^*R\), the eigenvalues different from zero of the operator
\[ \frac{A-A^*}{2i} \]
are eigenvalues of the operator \(T^{1/2}JT^{1/2}\) and therefore satisfy condition (1). From the listed properties of the operator \(A\) it follows that it admits a triangular representation \({}^{6}\). By a special choice of the operator \(A_1\), this representation can be written in the form
\[ A=\int_0^1 \varphi(x)\,dE(x)+2i\int_0^1 E(x)K\,dE(x) \quad \left(K=\frac{A-A^*}{2i}\right), \tag{8} \]
where \(\varphi(x)\) is a left-continuous nondecreasing function, and \(E(x)\) is such an absolutely continuous orthogonal resolution of the identity that the function \(R^*E(x)R\) is strictly increasing. The spectrum \(\mathfrak R\) of the operator (8) coincides with the set of singular points of the function \(W(\lambda)\) and with the closure of the set of values of the function \(\varphi(x)\) \((0<x\leqslant 1)\), and the integral sums
\[ A_\Delta=\sum_{k=1}^n \varphi(\xi_k)\Delta E_k +2i\sum_{r<s}\Delta E_rK\Delta E_s \]
\[ (0=x_0<x_1<\cdots<x_n=1;\qquad x_{k-1}<\xi_k\leqslant x_k;\quad \Delta E_k=E(x_k)-E(x_{k-1})) \]
converge in norm to \(A\) in the sense of S. O. Shatunovskii.
If \(\lambda\bar{\in}\mathfrak R\), then
\[ \left(\sum_{r<s}\frac{\Delta E_rK\Delta E_s}{\varphi(\xi_s)-\lambda}\right)^n=0 \]
and, consequently,
\[ (A_\Delta-\lambda E)^{-1} =\sum_{s=1}^n\frac{\Delta E_s}{\varphi(\xi_s)-\lambda} \left(E+2i\sum_{r<s}\frac{\Delta E_rK\Delta E_s}{\varphi(\xi_s)-\lambda}\right)^{-1} = \]
\[ =-\sum_{r=1}^n\frac{\Delta E_r}{\lambda-\varphi(\xi_r)} -2i\sum_{r<s}\frac{\Delta E_r}{\lambda-\varphi(\xi_r)} RJR^*\frac{\Delta E_s}{\lambda-\varphi(\xi_s)} - \]
\[ -(2i)^2\sum_{r<s<t} \frac{\Delta E_r}{\lambda-\varphi(\xi_r)}RJR^* \frac{\Delta E_s}{\lambda-\varphi(\xi_s)}RJR^* \frac{\Delta E_t}{\lambda-\varphi(\xi_t)} -\cdots \]
\[ \cdots-(2i)^{n-1} \frac{\Delta E_1}{\lambda-\varphi(\xi_1)}RJR^* \frac{\Delta E_2}{\lambda-\varphi(\xi_2)}RJR^*\cdots \frac{\Delta E_n}{\lambda-\varphi(\xi_n)}. \]
Thus,
\[ E-2iR^*(A_\Delta-\lambda E)^{-1}RJ = \prod_{k=1}^n \left( E+2i\,\frac{R^*\Delta E_kRJ}{\lambda-\varphi(\xi_k)} \right). \tag{9} \]
Since
\[ W(\lambda)=E-2iR^*(A-\lambda E)^{-1}RJ, \]
then, putting \(F(x)=R^*E(x)R\) and passing to the limit in equality (9), we obtain formula (2).
Theorem 2 is proved analogously. The only difference is that the operator \(A_0\) in this case is completely continuous, since its imaginary part is completely continuous and its spectrum consists of only one point, \(0\). In connection
With this, condition (1) becomes superfluous, and (7) and (8) are replaced by the formula
\[ A=2i\int_0^1 E(x)K\,dE(x)\qquad \left(K=\frac{A-A^*}{2i}\right). \tag{10} \]
3. Let us also note the following relations. Under the conditions of Theorem 1 there exists the integral
\[ W(x,y,\lambda)=\int_x^y\left(E+\frac{2i}{\lambda-\varphi(t)}\,dF(t)J\right)\qquad (x<y), \]
which, for arbitrary fixed \(x\) and \(y\), satisfies conditions I, II, III, and (1). The function \(W(x,y,\lambda)\) satisfies the integral equation
\[ W(x,y,\lambda)=E+2i\int_x^y \frac{W(x,t,\lambda)\,dF(t)J}{\lambda-\varphi(t)} \]
and expands into a norm-convergent series
\[ \begin{aligned} W(x,y,\lambda)=E &+2i\int_x^y \frac{dF(t)J}{\lambda-\varphi(t)} +(2i)^2\int_x^y\int_x^{x_1} \frac{dF(t)J}{\lambda-\varphi(t)} \frac{dF(x_1)J}{\lambda-\varphi(x_1)} \\ &+(2i)^3\int_x^y\int_x^{x_2}\int_x^{x_1} \frac{dF(t)J}{\lambda-\varphi(t)} \frac{dF(x_1)J}{\lambda-\varphi(x_1)} \frac{dF(x_2)J}{\lambda-\varphi(x_2)} +\cdots . \end{aligned} \]
Moreover,
\[ W(x,y,\lambda)W(y,z,\lambda)=W(x,z,\lambda)\qquad (x<y<z), \]
\[ W(x,y,\lambda)JW^*(x,y,\mu)-J = 2i(\overline{\mu}-\lambda) \int_x^y \frac{W(x,t,\lambda)\,dF(t)\,W^*(x,t,\mu)} {(\lambda-\varphi(t))(\overline{\mu}-\varphi(t))}. \]
Analogous relations hold under the conditions of Theorem 2.
Odessa State Pedagogical Institute
named after K. D. Ushinsky
Received
9 I 1961
CITED LITERATURE
- V. P. Potapov, DAN, 72, No. 5, 849 (1950).
- V. P. Potapov, Tr. Moscow Math. Soc., 4, 125 (1955).
- M. S. Livshits, Mat. Sbornik, 34 (76), 145 (1954).
- Yu. P. Ginzburg, DAN, 117, No. 2, 171 (1957).
- N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space, Moscow–Leningrad, 1950.
- M. S. Brodskii, DAN, 133, No. 6, 1271 (1960).
- M. S. Brodskii, UMN, 16, issue 1 (97) (1961).