MATHEMATICS

I. Ts. GOKHBERG and M. G. KREIN

ON THE THEORY OF TRIANGULAR REPRESENTATIONS OF NON-SELFADJOINT OPERATORS

(Presented by Academician A. N. Kolmogorov on 23 XI 1960)

1. In what follows, (\mathfrak H) denotes a separable Hilbert space; (\mathfrak R) is the normed ring of all linear bounded operators acting in (\mathfrak H); (\mathfrak S_\infty) is the Banach space of all completely continuous operators with the usual norm
[
|A|\infty=\sup(|Af|/|f|)\quad (A\in\mathfrak S_\infty).
]

Let (A\in\mathfrak S_\infty); by (s_n(A)) ((n=1,2,\ldots)) we shall denote the eigenvalues, numbered in decreasing order with regard to their multiplicities, of the nonnegative operator ((A^*A)^{1/2}).

As is known ((^1)), the set (\mathfrak S_p) ((1\le p<\infty)) of all operators (A\in\mathfrak S_\infty) for which
[
\sum_n s_n^p(A)<\infty
]
forms a Banach space with norm
[
|A|_p=\left(\sum_n s_n^p(A)\right)^{1/p}.
]

Let (\Pi={\pi_n}1^\infty) be an arbitrary nonincreasing sequence of positive numbers such that (\lim \pi_n=0) and (\sum_n \pi_n=\infty). Then the sets (\mathfrak S\Pi) and (\mathfrak S_\pi) of all operators (A(\in\mathfrak S_\infty)) for which, respectively,
[
|A|\Pi=\sup_n\left(\sum_1^n s_k(A)\sum_1^n \pi_k\right)<\infty,\qquad
|A|\pi=\sum_n \pi_n s_n(A)<\infty,
]
form Banach spaces with norms (|A|\Pi) and (|A|\pi).

Denote by (\mathfrak S_\Pi^{(0)}) the closure in (\mathfrak S_\Pi) of the lineal (\mathfrak R) of all finite-dimensional operators. It turns out that always
[
\mathfrak S_\Pi^{(0)}\ne \mathfrak S_\Pi.
]

All the introduced spaces (\mathfrak S) of completely continuous operators are two-sided self-adjoint ideals in (\mathfrak R).

An important case of the space (\mathfrak S_\pi) is the space (\mathfrak S_\omega), where
[
\Omega={(2n-1)^{-1}}1^\infty.
]
This space was introduced for consideration by V. I. Matsaev, who clarified its essential role in various questions. The introduction of this space served as an impetus for the consideration of the general spaces (\mathfrak S\Pi), (\mathfrak S_\Pi^{(0)}), and (\mathfrak S_\pi), and for establishing the connection between them.

Theorem 1. The general form of a linear continuous functional (F(X)) in (\mathfrak S), where (\mathfrak S) is one of the spaces (\mathfrak S_1), (\mathfrak S_p) ((1<p<\infty)), (\mathfrak S\Pi^{(0)}), (\mathfrak S_\pi), is given by the formula (F(X)=\operatorname{Sp}(A^X)), where (A) is an arbitrary operator from the space (\mathfrak S^), coinciding respectively with (\mathfrak R), (\mathfrak S_q) ((q^{-1}+p^{-1}=1)), (\mathfrak S_\pi), (\mathfrak S_\Pi), and moreover_
[
|F|=\sup_{X\in\mathfrak S}(|F(X)|/|X|{\mathfrak S})=|A|.
]

Thus the spaces (\mathfrak R), (\mathfrak S_q) ((q^{-1}+p^{-1}=1)), (\mathfrak S_\pi), (\mathfrak S_\Pi) are equivalent to the spaces conjugate respectively to the spaces (\mathfrak S_1), (\mathfrak S_p) ((1<p<\infty)), (\mathfrak S_\Pi^{(0)}), (\mathfrak S_\pi).

What is new in this theorem is only the assertion concerning the spaces (\mathfrak S_\Pi), (\mathfrak S_\Pi^{(0)}), and (\mathfrak S_\pi); the remaining assertions are known from the work of J. von Neumann and R. Schatten ((^1)).

Let (\mathfrak S) be one of the spaces introduced; then the set (\mathfrak G) of all self-adjoint operators from (\mathfrak S) is a real subspace of the space (\mathfrak S). Corresponding to any space (\hat{\mathfrak S}) there will be the corresponding space (\hat{\mathfrak S}^{}). In the space (\hat{\mathfrak S}) we introduce a new norm (\lvert H\rvert_{\mathfrak S;K}), topologically equivalent to the original one, by setting
[
\lvert H\rvert_{\mathfrak S;K}=\max(\lvert H^{+}\rvert_{\mathfrak S},\lvert H^{-}\rvert_{\mathfrak S}),
]
where (H^{+}, H^{-}) are mutually orthogonal nonnegative operators whose difference is (H). To this norm in the adjoint space there will correspond the norm
[
\lvert H\rvert_{\mathfrak S^{};K^{}}=\lvert H^{+}\rvert_{\mathfrak S^{}}+\lvert H^{-}\rvert_{\mathfrak S^{*}}.
]

1. A closed (in the sense of strong convergence) set of orthogonal projectors (\mathfrak P={P}) will be called a chain if it contains the projectors (0) and (I), and for any two projectors (P_1,P_2\in\mathfrak P) either (P_1<P_2), or (P_2<P_1). A pair of projectors ((P_1,P_2)) ((P_1<P_2;\; P_{1,2}\in\mathfrak P)) is called a gap of the chain (\mathfrak P) if for every projector (P\in\mathfrak P) either (P\leq P_1), or (P\geq P_2).

The dimension of the subspace ((P_2-P_1)\mathfrak H) of values of the projector (P_2-P_1) is called the dimension of the gap ((P_1,P_2)). A chain (\mathfrak P) having no gaps is called continuous.

A chain (\mathfrak P) is called maximal if it is not a proper part of any other chain. A maximal chain is characterized by the fact that either it is continuous, or every one of its gaps is one-dimensional.

We shall say that a chain (\mathfrak P) is a proper chain of an operator (A) if all subspaces (P\mathfrak H) are invariant with respect to (A), i.e. (PAP=AP) ((P\in\mathfrak P)).

Every operator (A\in\mathfrak S_\infty) possesses at least one proper maximal chain (see ((^{2-4}))).

A partition (\sigma) of the chain (\mathfrak P) will mean any chain consisting of a finite number of projectors
[
P_0<P_1<\cdots<P_n
]
from (\mathfrak P).

Let (F(P)) be an arbitrary operator-function, defined on the chain (\mathfrak P) and taking values in (\mathfrak S_\infty). If, for some (A\in\mathfrak S_\alpha), for every (\varepsilon>0) there exists a partition (\sigma_\varepsilon) of the chain (\mathfrak P) such that for all partitions (\sigma={P_j}0^n\supset\sigma\varepsilon)
[
\left|\sum_{j=1}^{n} F(Q_j)(P_j-P_{j-1})-A\right|<\varepsilon,
]
whatever the operators (Q_j\in\mathfrak P) satisfying the inequalities
[
P_{j-1}\leq Q_j\leq P_j \quad (j=1,2,\ldots,n)
]
may be, then we shall write
[
A=\int_{\mathfrak P} F(P)\,dP
]
and shall say that the integral on the right converges.

Every Volterra operator (Y) (i.e. every completely continuous operator (Y) with the single spectral point (\lambda=0)) admits a representation convergent in the norm of (\mathfrak S_\infty):
[
Y=2i\int_{\mathfrak P} PX\,dP,
\tag{1}
]
where (X=Y_J=(Y-Y^{})/2i) is the imaginary Hermitian component of the operator (Y), and (\mathfrak P) is an arbitrary maximal proper chain of the operator (Y).

* This proposition follows from a theorem of M. S. Brodskii ((^4)), who, instead of the definition of the integral (1) according to Schatten accepted here, proceeded from a definition connected with a preliminary parametrization of the chain (\mathfrak P).

Conversely (cf. (({}^{4,5}))), if for some (X=X^*\in \mathfrak S_\infty) the integral (1) converges in the norm of (\mathfrak S_\infty), then (Y) is the unique Volterra operator possessing the proper chain (\mathfrak P) and imaginary component (Y_J=X).

A necessary condition for the convergence of the integral (1) is the condition

[
(P-Q)\times(P-Q)=0.
\tag{2}
]

where ((P,Q)) is an arbitrary discontinuity of the chain (\mathfrak P).

Let (\mathfrak P) be an arbitrary continuous chain. Consider the operators

[
\mathcal T(X)=\mathcal T(X;\mathfrak P)=\int_{\mathfrak P} P X\,dP,\qquad
\mathcal S(X)=\mathcal S(X;\mathfrak P)=i\int_{\mathfrak P}(P X\,dP-dP\,X P).
]

Both operators are closed in (\mathfrak S_\infty) and have the same domain of definition (\mathfrak D).

For all (X\in\mathfrak D), (\mathcal T(\mathcal T(X))=X) and (\mathcal S(\mathcal S(X))=X).

Theorem 2. Let (\mathfrak S_{\mathrm I}) be one of the spaces (\mathfrak S_p) ((1<p<\infty)), (\mathfrak S_{\Pi}^{(0)}), (\mathfrak S_\pi), and let (\mathfrak S_{\Pi}) be one of the spaces (\mathfrak S_p) ((1<p\leq\infty)), (\mathfrak S_{\mathrm I}^{(0)}). If the space (\mathfrak S_{\mathrm I}\subset\mathfrak D) and is mapped by the operator (\mathcal T) continuously into the space (\mathfrak S_{\Pi}), then (\mathfrak S_{\Pi}^\subset\mathfrak D) and is mapped continuously by the operator (\mathcal T) into the space (\mathfrak S_{\mathrm I}^). Conversely, if the space (\mathfrak S_{\Pi}^\subset\mathfrak D) is mapped by the operator (\mathcal T) continuously into the space (\mathfrak S_{\mathrm I}^) and the additional condition (\mathcal T(\mathfrak K)\subset\mathfrak S_{\Pi}) is satisfied, then (\mathfrak S_{\mathrm I}\subset\mathfrak D) and is mapped continuously by the operator (\mathcal T) into the space (\mathfrak S_{\Pi}).

In both cases, of the two operators (\mathcal T(\mathfrak S_{\mathrm I}\to\mathfrak S_{\Pi})) and (\mathcal T(\mathfrak S_{\Pi}^\to\mathfrak S_{\mathrm I}^)), generated by the operator (\mathcal T) under these mappings, the second is adjoint to the first.

In the case (\mathfrak S_{\mathrm I}=\mathfrak S_1), the theorem remains valid if in its formulation the space (\mathfrak S_{\mathrm I}^*) is replaced by (\mathfrak S_\infty).

Exactly the same theorem holds for the operator (\mathcal S).

1. Almost without changing the arguments from Theorem 2 of (({}^{5})), one obtains the following refinement of it:

Theorem 3. Let (A) be a Volterra operator with (A_J\in\mathfrak S_1); then

[
|A|{\Omega}\leq \frac{4}{\pi}|A_J|_1,\qquad
|A_R||A_J|_1}\leq \frac{2}{\pi
\quad
\left(A_R=\frac12(A+A^*)\right).
\tag{3}
]

These estimates are sharp and are attained when (A_J) is a one-dimensional operator.

Let us add that for any self-adjoint operator (H\in\mathfrak S_1) and arbitrary (\varepsilon>0) there exists a Volterra operator (A) with (A_J=H) such that

[
|A|{\Omega}\geq \frac{4}{\pi}|A_J|_1-\varepsilon,\qquad
|A_R||A_J|_1-\varepsilon.}\geq \frac{2}{\pi
]

If the operator (H=H^*\in\mathfrak S_1) ((H\in\mathfrak S_\infty)), then there always exists a Volterra operator (A\in\mathfrak S_\Omega) with (A_J=H).

On the basis of the theorem of Hardy—Littlewood—Pólya—Weyl (({}^{6,7})), it follows from relations (3) that

[
\sum_{j=1}^{n} f(s_j(A))\leq
\sum_{j=1}^{n} f\left(\frac{4|A_J|_1}{\pi(2j-1)}\right)
\qquad (n=1,2,\ldots);
\tag{4}
]

[
\sum_{j=1}^{n} f(s_j(A_R^\pm))\leq
\sum_{j=1}^{n} f\left(\frac{2|A_J|_1}{\pi(2j-1)}\right)
\qquad (n=1,2,\ldots),
\tag{5}
]

where (f(t)) ((0 \le t < \infty)) is an arbitrary positive convex function for which (f(0)=0). If (A_J) is one-dimensional, equality holds in all relations (4) and (5).

Let us note, finally, that if (X \in \mathfrak S_1), and (\mathfrak P) is an arbitrary chain satisfying the necessary condition (2), then the exact estimate
(|\mathcal T(X;\mathfrak P)|_\Omega \le |X|_1) holds.

1. The combination of Theorems 1, 2, and 3 leads to an important result of V. I. Macaev, according to which, for every self-adjoint (and hence also non-self-adjoint) operator (X \in \mathfrak S_\infty) and any chain (\mathfrak P) satisfying condition (2), the integral (1) converges to some operator (Y \;(\in \mathfrak S_\infty))* and the exact estimate holds

[
|\mathfrak S(X)|\infty = |Y_R|\infty \le
\frac{2}{\pi}\sum_{j=1}^{\infty}
\frac{s_j(X^+) + s_j(X^-)}{2j-1}
\qquad (X=X^* \in \mathfrak S_\omega).
]

This result was obtained by V. I. Macaev by another method. The inequality
(|\mathcal T(X;\mathfrak P)|_\Omega \le |X|_1), on the basis of Theorem 2, makes it possible to supplement V. I. Macaev’s result with the following exact estimate

[
|\mathcal T(X)|\infty < |X|\omega
\qquad (X \in \mathfrak S_\omega).
]

Moldavian Branch
of the Academy of Sciences of the USSR

Odessa Civil Engineering Institute

Received
23 XI 1960

CITED LITERATURE

1. R. Schatten, A Theory of Cross-Spaces, Princeton, 1950.
2. H. Ароншайн, Р. Смит, Математика, Сборн. пер., 2, No. 1, 1958.
3. Л. А. Сахнович, Изв. Высших учебн. завед., Математика, No. 1 (8) (1959).
4. М. С. Бродский, УМН, 16, no. 1 (1961).
5. И. Ц. Гохберг, М. Г. Крейн, ДАН, 128, No. 2 (1959).
6. H. Weyl, Proc. Nat. Acad. Sci. USA, 35, 408 (1949).
7. G. Polya, Proc. Nat. Acad. Sci. USA, 36, 49 (1950).

* Prior to this, convergence of the integral (1) had been established by M. S. Brodskii (⁴) for the class (\mathfrak S_1); by the authors first for the class (\mathfrak S_2) (⁵), and then for all classes (\mathfrak S_p) ((0