MATHEMATICS

S. G. KREIN

ON AN INTERPOLATION THEOREM IN OPERATOR THEORY

(Presented by Academician A. N. Kolmogorov, 10 X 1959)

In the present theorem we give one general theorem of the theory of operators in Banach spaces, which contains, as special cases, a number of facts from various branches of functional analysis.

1. Let \(E\) be a Banach space, and \(M\) a certain linear set on which a family of linear operators \(T(z)\) is defined, acting from \(M\) into \(E\) and depending on the complex parameter \(z\).

Suppose the following conditions are satisfied:

A. For every \(x \in M\), the function \(T(z)x\) is an entire analytic function of \(z\) with values in the space \(E\) \((^1)\), not identically equal to zero.

B. The function \(\|T(z)x\|_E\) is bounded on every straight line parallel to the imaginary axis.

For each real \(\alpha\) introduce on the set \(M\) the norm

\[ \|x\|_\alpha=\sup_{-\infty<\tau<\infty}\|T(\alpha+i\tau)x\|_E \tag{1} \]

and complete the resulting normed space to the Banach space \(E_\alpha\). The family of Banach spaces \(E_\alpha\) \((-\infty<\alpha<\infty)\) will be called an analytic scale of spaces. Let us indicate an important property of the norm (1). It is not difficult to verify that the norm of the analytic function \(T(z)x\) is a logarithmically subharmonic function in the entire complex plane. By virtue of the theorem on three lines for a logarithmically subharmonic function, \(\|x\|_\alpha\) will be a logarithmically convex function of \(\alpha\), i.e., the following inequality will hold: for \(\alpha \leqslant \beta \leqslant \gamma\),

\[ \|x\|_\beta \leqslant \|x\|_\alpha^{\frac{\gamma-\beta}{\gamma-\alpha}} \|x\|_\gamma^{\frac{\beta-\alpha}{\gamma-\alpha}} \quad (x\in M). \tag{2} \]

From inequality (2) it follows that

\[ \|x\|_\beta \leqslant \frac{\gamma-\beta}{\gamma-\alpha}\, \varepsilon^{-(\gamma-\beta)}\|x\|_\alpha + \frac{\beta-\alpha}{\gamma-\alpha}\, \varepsilon^{\beta-\alpha}\|x\|_\gamma \quad (\varepsilon>0). \tag{3} \]

Conversely, if (3) is satisfied for every \(\varepsilon>0\), then (2) follows from it.

1. Example 1. An important example for us of an analytic scale of spaces will be the following: the space \(E=H\) is Hilbert; \(A\) is a positive self-adjoint operator acting in \(H\); \(M\) is an everywhere dense set in \(H\), consisting of elements on which all powers of the operator \(A\) are defined; the family of operators \(T(z)=A^z\). We shall denote the corresponding scale of spaces by \(H_A\), and the spaces themselves by \(H_\alpha\).

It is obvious that \(H_0=H\). If \(A\) is an unbounded positive-definite operator, then for \(\alpha>0\) one may identify \(H_\alpha\) with the domain of definition \(D(A^\alpha)\) of the power \(A^\alpha\) of the operator \(A\). Inequality (2) in this case coincides with the known inequality for moments \((^2)\).

Example 2. As a concrete example of the preceding construction, consider the Hilbert space \(H\) of complex-valued functions,

defined in \(n\)-dimensional space, with summable square of the modulus. For \(f(P)\in H\) denote by \(\tilde f(Q)\) its Fourier transform. Denote by \(A\) the operator which assigns to a function \(f(P)\) the function \(Af(P)\), whose Fourier transform has the form
\[ \widetilde{Af}(Q)=(1+|Q|)\tilde f(Q). \]
With its natural domain of definition this operator will be self-adjoint. The scale of spaces obtained by the scheme described above will be denoted by \(\{W_2^\alpha\}\). For \(Q\) equal to a positive integer \(l\), the space \(W_2^l\) will coincide with the corresponding Sobolev space \((^3)\), and for positive \(\alpha\), with the space of Slobodetskii \((^4)\).

Example 3. Consider continuous functions in the closure \(\overline G\) of an \(n\)-dimensional domain \(G\). The set \(M\) consists of functions \(f(P)\) that are equal to zero in a neighborhood (each in its own) of a certain manifold \(\Omega\), lying in \(\overline G\) and of smaller dimension (for example, \(\Omega\) is a point or \(\Omega\) is the boundary of the domain). On \(M\) we define the operators \(T(z)f(P)=f(P)/r^z\), where \(r\) is the distance from the point \(P\) to the manifold \(\Omega\), regarded as operators acting from \(M\) into the space \(E=L_p(G)\). The analytic scale of spaces constructed from these operators will be denoted by \(\{L_{p,\alpha}(G,\Omega)\}\) or \(\{L_{p,\alpha}\}\).

3. Generalizations. Such classical scales of spaces as spaces of functions satisfying Hölder conditions, or the spaces \(L_p\), do not fall under the concept of an analytic scale. The concept of an analytic scale can be extended. First of all, one may abandon the requirement that the function \(T(z)x\) be entire, and assume that it is analytic inside the strip \(\alpha_0<\operatorname{Re} z<\beta_0\). Under such a definition the Hölder classes \(C_\alpha\), with the natural norm, will form an analytic scale for \(0<\alpha<1\). To construct the scale of spaces \(L_p\), one should abandon the linearity of the operators \(T(z)\).

Example 4. Consider measurable functions in an \(n\)-dimensional domain \(G\). Let \(M\) denote the set of functions taking only a finite number of different values. Let \(E\) be the space \(L_1(G)\). Let \(T(z)f=|f|^z\). It is obvious that \(T(z)f\), for \(f\in M\), will be an entire analytic function with values in \(L_1(G)\). The function
\[ \varphi(t)=\sup_{-\infty<\tau<\infty}\bigl\|\,|f|^{t+i\tau}\,\bigr\|_{L_1} =\bigl\|\,|f|^t\,\bigr\|_{L_1}. \]
is logarithmically convex, but does not have the properties of a norm. The function
\[ [\varphi(1/\alpha)]^\alpha =\bigl\|\,|f|^{1/\alpha}\,\bigr\|_{L_1}^{\alpha} =\|f\|_{L_{1/\alpha}} \]
is logarithmically convex as a function of \(\alpha\) and has the properties of a norm for \(0\leqslant \alpha\leqslant 1\).

Thus, the scale \(L_p\) is constructed from the nonlinear operator \(T(z)\), analytic in \(z\), with a subsequent change of parameter.

4. Group property. In Examples 1–3 the operators \(T(z)\) form a commutative group, and therefore
\[ \|T(\beta+i\sigma)x\|_\alpha =\sup_{-\infty<\tau<\infty}\|T(\alpha+i\tau)T(\beta+i\sigma)x\|_E \]
\[ =\sup_{-\infty<\tau<\infty}\|T(\alpha+\beta+i(\tau+\sigma))x\|_E=\|x\|_{\alpha+\beta}. \]

In connection with this we introduce the following condition:

C. The set \(M\subset E\) is invariant with respect to the operators \(T(z)\). The operator \(T(0)\) is the identity. For every \(x\in M\) the function \(T(z)x\) is analytic in every space \(E_\alpha\), and
\[ \|T(\beta+i\sigma)x\|_\alpha\leqslant \|x\|_{\alpha+\beta}. \tag{4} \]

In Examples 1–3, condition C is satisfied; for Example 4 one can write an analogous condition, in which \(\alpha+\beta\) will be replaced by \(\alpha\beta\).

1. The conjugate scale. Let two analytic scales \(\{E_\alpha\}\) and \(\{E'_\alpha\}\) \((-\infty<\alpha<\infty)\) be given, constructed respectively on the sets \(M\) and \(M'\). We shall say that the scale \(\{E'_\alpha\}\) is conjugate to the scale \(\{E_\alpha\}\) if there exist a bilinear functional \((x,u)\), defined for \(x\in M\) and \(u\in M'\), and a linear correspondence \(\alpha \longleftrightarrow \alpha^*\) such that

\[ \|x\|_{E_\alpha}=\sup_{u\in M'}\frac{|(x,u)|}{\|u\|_{E'_{\alpha^*}}}. \tag{5} \]

It follows from (5) that each space \(E'_{\alpha^*}\) may be regarded as embedded in the space \(E_\alpha^*\), conjugate to \(E_\alpha\), and moreover \(\|u\|_{E_\alpha^*}\leq \|u\|_{E'_{\alpha^*}}\).

In Examples 1, 2, and 4 the scales are conjugate to themselves, with \(\alpha^*=-\alpha\) in Examples 1 and 2, and \(\alpha^*=1-\alpha\) in Example 4. In Example 3, the scale conjugate to \(\{L_{p,\alpha}\}\) is the scale \(\{L_{-p',-\alpha}\}\) \(\left(\frac1{p'}+\frac1p=1\right)\).

1. Interpolation theorem 1. Let \(\{E_\alpha\}\) and \(\{E_{\bar\alpha}\}\) be two analytic scales, and suppose there exists a scale \(\{E'_\alpha\}\) conjugate to \(\{E_{\bar\alpha}\}\). Assume that the scales \(\{E_\alpha\}\) and \(\{E'_\alpha\}\) satisfy condition C. Let an operator \(Q\) be defined on the set \(M\) corresponding to the scale \(\{E_\alpha\}\), such that for some \(\alpha,\beta\) and \(\bar\alpha,\bar\beta\)

\[ \|Qx\|_{\bar\alpha}\leq K_1\|x\|_\alpha,\qquad \|Qx\|_{\bar\beta}\leq K_2\|x\|_\beta \quad (x\in M). \tag{6} \]

Denote \(\alpha(\mu)=\mu\beta+(1-\mu)\alpha,\quad \bar\alpha(\mu)=\mu\bar\beta+(1-\mu)\bar\alpha.\) Then

\[ \|Qx\|_{\bar\alpha(\mu)}\leq K_1^{1-\mu}K_2^\mu\|x\|_{\alpha(\mu)}. \tag{7} \]

The method of proof of the theorem is close to the method of proof of M. Riesz’s theorem proposed by Calderón and Zygmund \((^5)\), and consists in applying the three-lines theorem to the analytic function

\[ \Phi(z)=(QT(z(\beta-\alpha))x,S^*(z(\bar\beta^*-\bar\alpha^*))y), \]

where \(T(z)\) and \(S^*(z)\) are the operators corresponding to the scales \(\{E_\alpha\}\) and \(\{E'_{\bar\alpha}\}\).

Remark. Analysis of the proof of the theorem shows that it admits a more general formulation. Without giving it, we note that this formulation is such that it includes the case when one of the scales, or both, are scales \(\{L_p\}\).

We give some consequences of Theorem 1.

Concrete interpolation theorems. In the case when both scales are scales \(\{L_p\}\), Theorem 1 coincides with the well-known theorem of M. Riesz.

Let now the scale \(\{E_\alpha\}=\{L_{p,\alpha}\}\), and \(\{E_{\bar\alpha}\}=\{L_{1/\bar\alpha}\}\). Then, for example, we arrive at the theorem:

Let the operator \(Q\) be a bounded operator acting in the space \(L_p\) with norm \(\|Q\|\). If the operator \(Q\) maps some space \(L_{p,\beta}\) \((\beta>0)\) into the space \(L_q\) \((q>p)\) and \(\|Qf\|_{L_q}\leq C\|f\|_{L_{p,\beta}}\), then it maps each space \(L_r\) \((p<r<q)\) into the space \(L_{p,\gamma}\), where \(\gamma=\dfrac{r-p}{q-p}\,\beta\) and \(\|Af\|_{L_r}\leq C^{\gamma/\beta}\|A\|^{1-\gamma/\beta}\|f\|_{L_{p,\gamma}}\).

Embedding theorems. We shall say that a Banach space \(E_2\) is embedded in the space \(E_1\) if \(E_2\subset E_1\) and \(\|x\|_{E_1}\leq C\|x\|_{E_2}\) \((x\in E_2)\). Applying Theorem 1 to the case when the operator \(Q\) is the identity operator, one can obtain a number of embedding theorems. From the embedding theorem of S. L. Sobolev \((^3)\) it follows that the space \(W_2^l\) for an integer \(l<n/2\) is embedded in the space \(L_{1/\bar\alpha}\), where \(\bar\alpha=\frac12-l/n\). On the other hand,

sides, the spaces \(W_2^0\) and \(L_2\) coincide. Consequently: for all positive \(\alpha<n/2\), the space \(W_2^\alpha\) is embedded in the space \(L_{\frac{2n}{\,n-\alpha\,}}\).

Similarly, using the results of \((^6)\), one can obtain, for example, the theorem:

For all positive \(\alpha<n/2\), the space \(W_2^\alpha\) is embedded in the space \(L_{2,\alpha}(G,P_0)\), where \(P_0\) is any point of the domain \(G\). The norm of a function in \(L_{2,\alpha}(G,P_0)\) is estimated in terms of the norm of the function in \(W_2^\alpha\), uniformly with respect to \(P_0\).

One can prove a number of similar assertions concerning embeddings of \(W_2^\alpha\) into spaces \(L_{p,\beta}(G,\Omega)\) with various \(p,\beta\) and manifolds \(\Omega\).

Theorems on fractional powers of operators. If in the general theorem one takes for the scale \(\{E_\alpha\}\) the scale \(H_A\) of Example 1, then we obtain:

Theorem 2. Let \(\{E_\alpha\}\) be an analytic scale whose conjugate satisfies condition C. Let \(E_0=H\), and let \(A\) be an unbounded positive self-adjoint operator acting in \(H\). If the domain of definition \(D(A)\) of the operator \(A\) is contained in the space \(E_{\bar\beta}\) and \(\|x\|_{E_{\bar\beta}}\le C\|Ax\|_H\), then the domain of definition \(D(A^\mu)\) of the operator \(A^\mu\) \((0\le\mu\le1)\) is contained in the space \(E_{\mu\bar\beta}\), and
\[ \|x\|_{E_{\mu\bar\beta}}\le C^\mu\|A^\mu x\|_H . \]

If for the scale \(\{E_\alpha\}\) one takes the scale \(L_p\), then we obtain the theorem of M. A. Krasnosel’skii with the refinement given in \((^6)\). If one sets \(E_\alpha=W_2^\alpha\), then we obtain a new theorem which, together with one of the embedding theorems formulated above, gives a refinement of results obtained by us with V. P. Glushko in \((^7)\). A new theorem is obtained likewise if one takes \(E_\alpha=L_{p,\alpha}\). Finally, if for the scale \(\{E_\alpha\}\) one takes the scale \(H_B\), constructed from some positive operator \(B\), then we arrive at the well-known inequality of E. Heinz \((^8)\): if \(\|Bx\|\le\|Ax\|\), then \(\|B^\mu x\|\le\|A^\mu x\|\). We note that from the interpolation theorem there also follows the following assertion of E. Heinz, with T. Kato’s refinement \((^9)\): if the operator \(Q\) is such that \(\|Qx\|\le\|Ax\|\) and \(\|Q^*x\|\le\|Bx\|\), then \(|(Qx,y)|\le\|A^\mu x\|\,\|B^{1-\mu}y\|\) \((0\le\mu\le1)\). Here the operator \(Q\) is regarded as an operator acting from the spaces of the scale \(H_A\) into the spaces of the scale \(H_B\). Inequalities (6) are then satisfied for the indices \(\bar\alpha=-1\), \(\bar\beta=0\), \(\alpha=0\), \(\beta=1\). The assertion of Heinz then follows from (7).

7. Complete continuity.

Theorem 3. If, under the conditions of Theorem 1, the operator \(Q\) is completely continuous as an operator from \(E_\alpha\) to \(E_{\bar\alpha}\), then it is completely continuous as an operator from \(E_{\alpha(\mu)}\) to \(E_{\gamma}\), where \(\gamma\) is any number between \(\bar\alpha\) and \(\alpha(\mu)\).

Theorem 3 follows directly from Theorem 1 and inequality (2).

8. Uniqueness.

Theorem 4. Suppose two Banach spaces \(F\) and \(G\) are given. If there exists an analytic scale \(\{E_\alpha\}\), whose conjugate has property C, such that \(E_0=F\) and \(E_1=G\), then such a scale on the interval \([0,1]\) is uniquely determined.

Received
9 X 1959

CITED LITERATURE

1. E. Hille, Functional Analysis and Semigroups, IL, 1951, p. 74.
2. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956, p. 58.
3. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950, p. 64.
4. L. N. Slobodetskii, DAN, 118, No. 2 (1958).
5. A. P. Calderon, A. Zygmund, Am. J. Math., 78, No. 2 (1956).
6. M. A. Krasnosel’skii, E. I. Pustyl’nik, DAN, 122, No. 6 (1958).
7. V. P. Glushko, S. G. Krein, DAN, 122, No. 6 (1958).
8. E. Heinz, Math. Ann., 123, 415 (1951).
9. T. Kato, Math. Ann., 125, 208 (1952).