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UDC 517.397.1:513.88.2+517.948.32
MATHEMATICS
M. Sh. Birman, M. Z. Solomyak
ON DOUBLE OPERATOR STIELTJES INTEGRALS
(Presented by Academician V. I. Smirnov on 3 V 1965)
1. In this paper we study operators in a Hilbert space \(\mathfrak H\) that are given by integrals of the form
\[ Q=\iint \varphi(\lambda,\mu)\, dF_\mu\, T\, dE_\lambda . \tag{1} \]
Here \(F_\mu, E_\lambda\) are any two orthogonal resolutions of the identity; \(T\) is a bounded operator in \(\mathfrak H\); \(\varphi(\lambda,\mu)\) is a complex function. Below, as usual, \(\mathfrak R\) denotes the ring of all bounded operators in \(\mathfrak H\); \(\mathfrak S_\infty\) is the ideal of completely continuous operators in \(\mathfrak H\); \(\mathfrak S_p\) \((1 \le p < \infty)\) is the ideal of all operators from \(\mathfrak S_\infty\) whose sequence of singular numbers belongs to \(l_p\). In particular, \(\mathfrak S_1\) is the class of nuclear operators; \(\mathfrak S_2\) is the Hilbert–Schmidt class.
Integrals of the form (1) constitute a useful apparatus in a number of questions of operator theory. Apparently they first appeared in the work \((^1)\) in connection with certain questions of the analytic theory of perturbations. In \((^1)\) the simplest condition for convergence of such integrals in \(\mathfrak R\) was indicated. In \((^{2,3})\) the significance of integrals (1) in the theory of perturbations on the continuous spectrum became clear. In particular, it became necessary to study these integrals in some of the classes \(\mathfrak S_p\).
Our aim is to study systematically the properties of the operator \(Q\) as a function of the behavior of the function \(\varphi(\lambda,\mu)\) and of the properties of the operator \(T\). In doing so we clarify the question of the sense in which the integrals (1) are to be understood, and establish relations among the various possible definitions of the integral. We shall regard the integral (1) as an operator on \(T\), acting from one of the classes \(\mathfrak S_p\) \((1 \le p \le \infty)\), \(\mathfrak R\) into another such class. (Following I. Ts. Gohberg and M. G. Krein \((^4)\), operators in spaces of operators will be called transformers. If a transformer \(\Phi\) is bounded as an operator from \(\mathfrak S_p\) to \(\mathfrak S_q\), then we shall call it a transformer of class \((\mathfrak S_p,\mathfrak S_q)\).)
We note that the study of the connection between the Hermitian components of a Volterra operator \((^{5-10})\) is related to the study of one special transformer of the form (1). Namely, the well-known formula
\[ Q=2i\int_0^1 E_\lambda T\, dE_\lambda , \]
which reconstructs the Volterra operator \(Q\) from its imaginary component \(T\), is written by means of the integral (1) for \(E_\lambda=F_\lambda\) and \(\varphi(\lambda,\mu)= i[\operatorname{sign}(\lambda-\mu)+1]\). It should be noted at once that the results of \((^{5-10})\) follow from our theorems only in the cases where \(T\in\mathfrak S_2\) or \(T\in\mathfrak S_1\). We also point out that some transformers studied in \((^4)\) can also be written in the form (1). There, however, somewhat different questions were considered. Finally, we note that multidimensional singular integrals with characteristic depending on the pole \((^{11})\) are essentially integrals of the form (1). In this case \(T=I\), and both spect-
real families \(F_\mu, E_\lambda\) depend on multidimensional points \(\mu\) and \(\lambda\). This question will be discussed in more detail elsewhere.
- Let us first consider the integral (1) for \(T \in \mathfrak S_2\). We shall use the fact that the class \(\mathfrak S_2\) is a Hilbert space with respect to the scalar product \(\langle T_1,T_2\rangle=\operatorname{Sp} T_2^*T_1\). The families of transformations \(\mathcal F_\mu,\mathcal E_\lambda\), defined by the formulas \(\mathcal F_\mu T=F_\mu T,\ \mathcal E_\lambda T=TE_\lambda\), are decompositions of the identity in \(\mathfrak S_2\). They obviously commute:
\[ \mathcal F_\mu\mathcal E_\lambda T=\mathcal E_\lambda\mathcal F_\mu T=F_\mu TE_\lambda . \]
The commuting families \(\mathcal F_\mu,\mathcal E_\lambda\) generate in the usual way \({}^{(12)}\) a spectral measure \(\mathcal G(e)\) in the plane. Moreover, if \(\varphi(\lambda,\mu)\) is \((\mathcal G)\)-measurable and bounded with respect to the measure \(\mathcal G(e)\), i.e.
\[ \|\varphi\|_G\equiv(\mathcal G)-\sup|\varphi(\lambda,\mu)|<\infty, \tag{2} \]
then the integral
\[ \Phi=\iint \varphi(\lambda,\mu)\,d\mathcal G(e) \tag{3} \]
defines a normal operator (transformation) in \(\mathfrak S_2\) with norm \(\|\varphi\|_G\).
Definition. For any \(T\in\mathfrak S_2\), the double operator integral (1) is called the value \(Q\) of the transformation (3) on the operator \(T\): \(Q=\Phi T\).
The validity of the following assertion is now evident.
Theorem 1. Let the \((\mathcal G)\)-measurable function \(\varphi(\lambda,\mu)\) satisfy condition (2). Then for every \(T\in\mathfrak S_2\) the operator integral (1) converges in the norm of \(\mathfrak S_2\) and defines a transformation \(\Phi\) of the class \((\mathfrak S_2,\mathfrak S_2)\). The set of transformations \(\Phi\) of the form (3) forms a commutative normed ring, isomorphic and isometric to the ring of \((\mathcal G)\)-measurable functions with norm (2). Moreover, to the adjoint transformation \(\Phi^*\) there corresponds the function \(\overline{\varphi(\lambda,\mu)}\).
Let \(E_\lambda,F_\mu\) correspond in \(\mathfrak H\) to self-adjoint operators \(A,B\). It is clear that the ring of transformations (3) coincides with the ring of bounded functions of the commuting (possibly unbounded) transformations of right multiplication by \(A\) and left multiplication by \(B\) in \(\mathfrak S_2\).
- In studying the integrals (1) in other classes we can use the definition of item 2 only for \(T\in\mathfrak S_p,\ 1\le p\le 2\). If the transformation \(\Phi\in(\mathfrak S_1,\mathfrak S_1)\) corresponding to the function \(\varphi(\lambda,\mu)\), then, obviously, the adjoint transformation \(\Phi^*\in(\mathfrak R,\mathfrak R)\). It is now simplest to define the integral (1) for \(T\in\mathfrak R\) by the formula \(Q=\Phi^*T\). It is clear that for \(T\in\mathfrak S_\infty\) one also has \(Q\in\mathfrak S_\infty\). In an analogous way one can define the integral (1) also in the case of the classes \(\mathfrak S_p,\ p>2\). Taking this definition as basic, we shall return to this question in item 4.
Theorem 2. Let the spectral family \(E_\lambda\) \((F_\mu)\) be constant outside a finite interval, and let the bounded Borel function \(\varphi(\lambda,\mu)\) satisfy the condition \(\operatorname{Lip}\alpha\) in the variable \(\lambda\) \((\mu)\), with a constant independent of \(\mu\) (of \(\lambda\)). If \(\alpha>1/2\), then the transformation \(\Phi\) defined by the integral (1) belongs to every class \((\mathfrak S_p,\mathfrak S_p)\), \(1\le p\le\infty\), and to the class \((\mathfrak R,\mathfrak R)\). If \(\alpha\le 1/2\), then \(\Phi\in(\mathfrak S_p,\mathfrak S_p)\), \(2(1+2\alpha)^{-1}<p<2(1-2\alpha)^{-1}\).
Remarks. 1) The case of an infinite interval of integration can be treated by means of a change of variable. 2) It can be shown that the results of Theorem 2 cannot be improved in terms of the Lipschitz classes. 3) For \(\alpha=1/2\) under the conditions of Theorem 2 one can additionally assert,* that \(\Phi\in(\mathfrak S_1,\mathfrak S_\omega)\) and \(\Phi\in(\mathfrak S_\omega,\mathfrak S_\infty)\). 4) The preceding assertion remains valid if the condition \(\varphi\in\operatorname{Lip}1/2\) is replaced by the condition of bounded variation of \(\varphi(\lambda,\mu)\) with respect to the variable \(\lambda\) \((\mu)\), uniformly in \(\mu\) \((\lambda)\). In this case there is no need to assume the interval of variation
* For the definition of the normed ideals \(\mathfrak S_\omega,\mathfrak S_\Omega\), introduced by V. I. Matsaev, I. Ts. Gohberg, and M. G. Krein, see, for example, in \({}^{(7)}\).
\(\lambda(\mu)\) is finite. 5) For \(\alpha > 1/2\) the transformer \(\Phi\) belongs to any class \((\mathfrak S,\mathfrak S)\), where \(\mathfrak S\) is an arbitrary minimal or maximal normed ideal in \(\mathfrak S_\infty\). This fact follows directly from the analogue of B. S. Mityagin’s interpolation theorem \((^{16})\) for ideals in \(\mathfrak S_\infty\).
- Under the hypotheses of Theorem 2, the integral (1) can be given meaning as the limit of integral sums. Namely, if one introduces the operator-valued function
\[ K(\lambda)=\int \varphi(\lambda,\mu)\,dF_\mu, \]
then the integral (1) can be written (for the time being formally) in the form of the repeated integral
\[ Q=\int K(\lambda)T\,dE_\lambda . \tag{4} \]
The convergence of the integral (4) is ensured by the following theorem.
Theorem 3. Let the bounded Borel function \(\varphi(\lambda,\mu)\) satisfy a \(\operatorname{Lip}\alpha\) condition in the variable \(\lambda\), with a constant independent of \(\mu\), and let the family \(E_\lambda\) be constant outside a finite interval. Then, for \(T\in\mathfrak R\) when \(\alpha>1/2\), and for \(T\in\mathfrak S_p,\ 2<p<2(1-2\alpha)^{-1}\), when \(\alpha\le 1/2\), the integral (4), understood as the limit in \(\mathfrak R\) of arbitrary integral sums of the form
\[ \sum_{k=0}^{n-1} K(\widetilde{\lambda}_k)T\,(E_{\lambda_{k+1}}-E_{\lambda_k}),\qquad \widetilde{\lambda}_k\in[\lambda_k,\lambda_{k+1}), \tag{5} \]
exists and coincides with the integral (1), understood in the sense of the definition in § 3. An analogous assertion is valid upon interchanging the roles of the variables \(\lambda\) and \(\mu\).
If both families \(E_\lambda,\ F_\mu\) are constant outside finite intervals and \(\varphi(\lambda,\mu)\in \operatorname{Lip}\alpha\) jointly in the variables \(\lambda,\mu\), then the double integral sums
\[ \sum_{k=0}^{m-1}\sum_{l=0}^{n-1} \varphi(\lambda_{kl},\mu_{kl})\,(F_{\mu_{k+1}}-F_{\mu_k})T(E_{\lambda_{l+1}}-E_{\lambda_l}) \tag{6} \]
converge in \(\mathfrak R\) to the integral (1).
Let us also note that, under the hypotheses of Theorem 3, for \(\alpha>1/2\) and \(T\in\mathfrak S_p\) \((1\le p\le\infty)\), the integral sums of the form (5), (6) converge to the integral (1) in the norm of \(\mathfrak S_p\).
The proof of Theorem 3 is based on certain results concerning the integration of additive set-functions not having bounded variation. In view of the lack of space, we give only the simplest assertion of this kind.
Theorem 4. Let \(f(\Delta)\) be an additive vector-valued function of half-open intervals \(\Delta\subset [a,b)\) with values in a Banach space \(\mathfrak X\), such that for some \(M,\gamma>0\), for any partition of \([a,b)\) into non-overlapping intervals \(\Delta_k=[\lambda_k,\lambda_{k+1})\), the inequality
\[ \sum_{k=0}^{n-1}\|f(\Delta_k)\|(\lambda_{k+1}-\lambda_k)^\gamma\le M \]
holds.
Furthermore, let \(K(\lambda)\) be an operator-valued function in \(\mathfrak X\) satisfying on \([a,b]\) the condition \(\operatorname{Lip}\alpha\) \((\alpha>\gamma)\) in the operator norm. Then the integral
\[ \int_a^b K(\lambda) f(d\lambda) \]
exists as a strong limit in \(\mathfrak X\) of Riemann–Stieltjes integral sums.
Theorem 4 may be regarded as a generalization of V. T. Kondurar’s theorem \({}^{(13)}\) on the integrability of a function \(f(t)\in \operatorname{Lip}\alpha\) with respect to a function \(g(t)\in \operatorname{Lip}\beta\) when \(\alpha+\beta>1\).
- For certain special classes of transformers the result of Theorem 2 can be sharpened.
Theorem 5. Let the derivative \(\psi'(\lambda)\) of the function \(\psi(\lambda)\) satisfy a \(\operatorname{Lip}\alpha\) condition for some \(\alpha>0\). Then the transformer
\[ \int_{a}^{b}\int_{a}^{b} [\psi(\lambda)-\psi(\mu)](\lambda-\mu)^{-1}\,dF_{\mu}T\,dE_{\lambda} \]
belongs to each of the classes \((\mathfrak R,\mathfrak R)\), \((\mathfrak S_p,\mathfrak S_p)\), \(1\le p\le \infty\).
A consequence of Theorem 5 is
Theorem 6. Let \(U,V\) be unitary operators in \(\mathfrak H\), and let \(\xi(t)\) be a function on the circle \(|t|=1\) whose derivative \(\xi'(t)\in \operatorname{Lip}\alpha\) \((\alpha>0)\). If \(V-U\in \mathfrak S_p\) \((1\le p\le \infty)\), then
\[ \xi(V)-\xi(U)\in \mathfrak S_p . \]
An analogous result is valid for functions of self-adjoint operators. The assertion of Theorem 6 for \(p=1\) is of definite interest for the abstract theory of scattering \({}^{(2,3)}\) and the theory of the spectral shift function \({}^{(14,15)}\). In particular, it is easy to show that, under the hypotheses of Theorem 6, the trace formula \({}^{(14)}\) remains valid for unitary operators.
Leningrad State University
named after A. A. Zhdanov
Received
29 IV 1965
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