UDC 519.4:517:513.88

MATHEMATICS

V. I. OVCHINNIKOV

SYMMETRIC SPACES OF MEASURABLE OPERATORS

(Presented by Academician M. A. Lavrent’ev on 26 VIII 1969)

We consider the space with measure on projections introduced by I. E. Segal \((^1)\), \(\Gamma=(H,\mathfrak A,m)\), i.e., a totality consisting of a Hilbert space \(H\), a Neumann algebra \(\mathfrak A\) of bounded operators acting in \(H\), and a nonnegative measure \(m\) defined on the orthoprojections from \(\mathfrak A\).

\(1^\circ\). Let \(A\) be an operator measurable with respect to the algebra \(\mathfrak A\); then the operators \(E_\lambda\) \((E_{\lambda-0}\ne E_\lambda)\) from the spectral decomposition of the identity for the operator \(|A|\) belong to the algebra \(\mathfrak A\). Denote by \(C_0(\Gamma)\) the totality of all operators for which the function \(m(I-E_\lambda)\) is finite for all \(\lambda>0\). In this case we shall call the function \(n_A(\lambda)=m(I-E_\lambda)\) the distribution function of the operator \(A\). We introduce into consideration the nonincreasing, left-continuous function \(s_A(\alpha)\) by the formula

\[ s_A(\alpha)=\inf\{\lambda;\ n_A(\lambda)<\alpha\}. \]

The values of the function \(s_A(\alpha)\) will be called the \(s\)-numbers of the operator \(A\). It follows from the definition that \(s_A(\alpha)\to 0\) as \(\alpha\to\infty\).

The minimax formula is valid

\[ s_A(\alpha)=\inf_{m\mathfrak R^\perp<\alpha}\ \sup_{f\in\mathfrak R\cap D(A)} \frac{\|Af\|}{\|f\|}, \tag{1} \]

where the measure of the subspace \(\mathfrak R^\perp\) is understood as the measure of the corresponding orthoprojector, if it belongs to the algebra \(\mathfrak A\), and as \(\infty\) otherwise.

For factors of finite type an analogue of formula (1) was established by F. J. Murray and J. von Neumann (see \((^2)\)). A result of J. Allahverdiev is generalized (see \((^3)\)): for any operator from \(C_0(\Gamma)\)

\[ s_A(\alpha)=\inf_K\|A-K\|, \]

where \(K\) runs through the set of all operators for which the measure \(\overline{R(K)}\) is less than \(\alpha\). From the two assertions just given, in the usual way, one derives a number of properties of \(s\)-numbers (see \((^3)\), Ch. I, § 2).

\(2^\circ\). For a positive operator \(A\in C_0(\Gamma)\) one constructs a self-adjoint operator \(\widetilde A\in n_A(A)\), where \(n_A(\lambda)\) is the distribution function of the operator \(A\), extended by zero at zero. If \(P_\alpha\) is the spectral decomposition of the identity for the operator \(\widetilde A\) and \(\widetilde E_\alpha=P_\alpha=P_{+0}\), then the equality

\[ A=\int_0^\infty s_A(\alpha)\,d\widetilde E_\alpha, \tag{2} \]

is valid, which we call the Schmidt decomposition of the operator \(A\).

In the set of all measurable operators one singles out the Banach space \(\mathscr L_1(\Gamma)\) (see \((^1)\)) of integrable operators, for which the integral or trace \(m(A)\) is defined. Moreover, \(\|A\|_{\mathscr L_1(\Gamma)}=m(|A|)\) and \(|m(A)|\le \|A\|_{\mathscr L_1(\Gamma)}\). It turns out that

\[ \|A\|_{\mathscr L_1(\Gamma)}=\int_0^\infty s_A(\alpha)\,d\alpha. \tag{3} \]

Using formulas (2) and (3), a number of integral inequalities for \(s\)-numbers are established.

Let \(\psi(\alpha)\) \((0 \leq \alpha < \infty)\) be a nondecreasing concave function and \(\psi(0)=0\); then

\[ \int_0^\infty s_{A+B}(\alpha)\,d\psi(\alpha) \leq \int_0^\infty s_A(\alpha)\,d\psi(\alpha) + \int_0^\infty s_B(\alpha)\,d\psi(\alpha), \]

\[ \int_0^\infty s_{A_1A_2\ldots A_n}(\alpha)\,d\psi(\alpha) \leq \int_0^\infty s_{A_1}(\alpha)s_{A_2}(\alpha)\ldots s_{A_n}(\alpha)\,d\psi(\alpha). \]

3°. A linear normed space \(E\) of measurable operators is called symmetric if, for any operator \(A\in E\) and measurable operator \(B\), it follows from the relation
\[ s_B(\alpha)\leq s_A(\alpha)\quad (0<\alpha<\infty) \]
that \(B\in E\) and \(\|B\|_E\leq \|A\|_E\).

The simplest examples of symmetric spaces are the spaces \(\mathscr L_p(\Gamma)\) \((1\leq p\leq \infty)\).

Theorem 1. From every sequence of operators in a symmetric space \(E\) that converges to zero, one can extract a subsequence that converges to zero almost everywhere.

There is a general method for constructing symmetric spaces, proposed in the scalar case by E. M. Semenov (see \((^4)\)).

Let \(\psi_t(\alpha)\) \((t\in T)\) be a family of nondecreasing concave functions on \([0,\infty)\), equal to zero at zero. Consider the family \(E_T(\Gamma)\) of all measurable operators for which

\[ \|A\|_{E_T(\Gamma)}=\sup_{t\in T}\int_0^\infty s_A(\alpha)\,d\psi_t(\alpha)<\infty . \tag{4} \]

Theorem 2. The space \(E_T(\Gamma)\), with respect to the norm (4), is a Banach symmetric space.

4°. For symmetric Banach spaces of measurable operators the embedding theorem is valid.

Theorem 3. If \(E\) is a Banach symmetric space, then
\[ \mathscr L_1(\Gamma)\cap \mathscr L_\infty(\Gamma)\subset E\subset \mathscr L_1(\Gamma)+\mathscr L_\infty(\Gamma), \]
and moreover
\[ \|A\|_{\mathscr L_1(\Gamma)+\mathscr L_\infty(\Gamma)} = \inf\{\|A_1\|_{\mathscr L_1(\Gamma)}+\|A_2\|_{\mathscr L_\infty(\Gamma)}\} = \int_0^1 s_A(\alpha)\,d\alpha, \tag{5} \]
where the infimum is taken over all representations \(A=A_1+A_2\) \((A_1\in \mathscr L_1(\Gamma),\ A_2\in \mathscr L_\infty(\Gamma)=\mathfrak A)\).

Equality (5) is a special case of the following assertion.

Theorem 4. If \(A\in C_0(\Gamma)\), then

\[ \int_0^t s_A(\alpha)\,d\alpha = \inf_{A=A_1+A_2} \{\|A_1\|_{\mathscr L_1(\Gamma)}+t\|A_2\|_{\mathscr L_\infty(\Gamma)}\}. \]

This assertion for the scalar case was obtained in \((^5,^6)\).

5°. Various interpolation theorems are generalized to symmetric spaces of measurable operators (see \((^7)\)).

Theorem 5. In order that the space \(E\) of measurable operators be an interpolation space between the spaces \(\mathscr L_1(\Gamma)\) and \(\mathscr L_\infty^0(\Gamma)=\mathscr L_\infty(\Gamma)\cap C_0(\Gamma)\), it is necessary and sufficient that, for every \(A\in E\), from the relation
\[ \int_0^t s_B(\alpha)\,d\alpha \leq \int_0^t s_A(\alpha)\,d\alpha \quad (t>0) \]
it follows that \(B\in E\).

We note that an analogue of Theorem 5 for the scalar case was obtained by A. P. Calderon \((^8)\), and for symmetrically normed ideals of completely continuous operators by G. I. Russo \((^9)\).

From this theorem it follows, in particular, that, up to an equivalent renorming, every interpolation space between \(\mathscr L_1(\Gamma)\) and \(\mathscr L_\infty^0(\Gamma)\) is symmetric. Further, it follows from Theorem 5 that the space \(E_T(\Gamma)\), which was discussed in Theorem 2, is always an interpolation space between \(\mathscr L_1(\Gamma)\) and \(\mathscr L_\infty(\Gamma)\).

Linear mappings acting in spaces of measurable operators will, following \((^3)\), be called transformers.

Theorem 6 (M. Riesz convexity theorem). Every transformer \(T\) acting continuously from \(\mathscr L_{1/\alpha_0}(\Gamma_0)\) to \(\mathscr L_{1/\beta_0}(\Gamma_1)\) and from \(\mathscr L_{1/\alpha_1}(\Gamma_0)\) to \(\mathscr L_{1/\beta_1}(\Gamma_1)\) acts continuously from \(\mathscr L_{1/\alpha_t}(\Gamma_0)\) to \(\mathscr L_{1/\beta_t}(\Gamma_1)\), and moreover
\[ \|T\|_{\mathscr L_{1/\alpha_t}\to\mathscr L_{1/\beta_t}} \le \|T\|_{\mathscr L_{1/\alpha_0}\to\mathscr L_{1/\beta_0}}^{\,1-t} \|T\|_{\mathscr L_{1/\alpha_1}\to\mathscr L_{1/\beta_1}}^{\,t}, \]
where
\[ \alpha_t=(1-t)\alpha_0+t\alpha_1,\qquad \beta_t=(1-t)\beta_0+t\beta_1\quad (0\le t\le 1). \]

6°. The method by which Theorems 5 and 6 were obtained makes it possible to carry over to symmetric spaces of measurable operators also other interpolation theorems proved for symmetric spaces of measurable functions. This method is based on the following construction.

Let \(\mathfrak a\) be a weakly closed subring of the algebra \(\mathfrak A\), containing the identity operator. We shall call the subring \(\mathfrak a\) proper if the aggregate \((H,\mathfrak a,m|_{\mathfrak a})=\gamma\), where \(m|_{\mathfrak a}\) is the restriction of the measure \(m\) to the orthoprojectors from \(\mathfrak a\), is a space with measure on the projectors.

For \(A\in \mathscr L_\infty(\Gamma)\), the expression \(m(A\cdot X)\) \((X\in \mathscr L_1(\gamma))\) will be a linear continuous functional on \(\mathscr L_1(\gamma)\) and, consequently, it is representable (see \((^1)\)) in the form
\[ m(A\cdot X)=m(Q(A)\cdot X), \]
where \(Q(A)\in \mathscr L_\infty(\gamma)\). Thus a projection transformer \(Q\) from \(\mathscr L_\infty(\Gamma)\) to \(\mathscr L_\infty(\gamma)\) is defined. It can be extended to a continuous transformer mapping \(\mathscr L_1(\Gamma)+\mathscr L_\infty(\Gamma)\) to \(\mathscr L_1(\gamma)+\mathscr L_\infty(\gamma)\).

Theorem 7. The transformer \(Q\) maps every space \(E_T(\Gamma)\) into \(E_T(\gamma)\) and has norm one.

The transformer \(Q\) makes it possible to pass from transformers acting in the spaces \(E_T(\Gamma)\) to transformers acting in the simpler spaces \(E_T(\gamma)\), and conversely. By choosing the subrings \(\mathfrak a\) to be commutative, we establish connections between interpolation theorems for operators acting in spaces of measurable functions and analogous theorems for transformers in spaces of measurable operators.

We note that the indicated method for the case of symmetrically normed ideals of completely continuous operators is close to the method described in the dissertation of M. Z. Solomyak \((^{10})\).

The author thanks S. G. Krein for his attention to the work and for his advice.

Voronezh State University
named after the Lenin Komsomol

Received
8 VIII 1969

REFERENCES

\(^1\) I. E. Segal, Ann. Math., 57, 401 (1953); Collection of translations, Matematika, 6, 1 (1962).
\(^2\) M. A. Naimark, Normed Rings, Nauka, 1968.
\(^3\) I. Ts. Gohberg, M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators, Nauka, 1965.
\(^4\) E. M. Semenov, Functional Analysis, 2, no. 2 (1968).
\(^5\) G. Petre, Rend. Sem. Mat. Fis. Milano, 34 (1964).
\(^6\) B. S. Mitjagin, Mat. sbornik, 66, 473 (1965).
\(^7\) E. Madzhenes, UMN, 21, no. 2 (128) (1966).
\(^8\) A. P. Calderon, Studia Math., 26, 273 (1966).
\(^9\) G. I. Russo, Functional Analysis, 3, no. 2 (1969).
\(^ {10}\) M. Z. Solomyak, Doctoral dissertation, Leningrad University, 1968.