Full Text
Reports of the Academy of Sciences of the USSR
1964, Volume 159, No. 5
MATHEMATICS
P. P. Zabrejko
On Some Properties of Linear Operators Acting in the Spaces \(\mathscr L_p\)
(Presented by Academician A. Yu. Ishlinskii on 12 VI 1964)
In recent years a number of new theorems have been established on the continuity and complete continuity of linear and, in particular, linear integral operators \((^{1-6})\). In these theorems an essential role is played by the property of compactness of operators with respect to measure. The author has undertaken an attempt at a detailed analysis of this property. It has turned out that the systematic use of the property of compactness with respect to measure for linear operators, in combination with the concept of regularity (see \((^7)\)), makes it possible to strengthen a number of known theorems and to establish some new assertions. The article also analyzes certain other properties of linear operators closely connected with the property of compactness with respect to measure.
Let a measure \(\mu\) be given on a set \(\Omega\). As usual, by \(\mathscr L_p\) \((0<p<\infty)\) we denote the class of measurable functions for which
\[ \|x\|_p=\left\{\int_\Omega |x(t)|^p\,d\mu(t)\right\}^{1/p}<\infty . \]
For \(p\geqslant 1\) the class \(\mathscr L_p\) is a Banach space; for \(p<1\) the class \(\mathscr L_p\) is not a Banach space, but it is a complete metric space with metric \(\rho(x,y)=\|x-y\|_p^p\). By \(\mathscr L_\infty\) we denote the Banach space of essentially bounded functions with norm
\[ \|x\|_\infty=\operatorname{vrai\,sup}_{t\in\Omega}|x(t)|. \]
By \(P_D\) we denote the operator of multiplication by the characteristic function of a set \(D\subset\Omega\); this operator is continuous in each of the spaces \(\mathscr L_p\).
We consider linear continuous operators acting from the space \(\mathscr L_p=\mathscr L_p(\Omega_1)\) into the space \(\mathscr L_q=\mathscr L_q(\Omega_2)\). Here \(\Omega_1\) and \(\Omega_2\) are two sets with finite and non-atomic measures \(\mu_1\) and \(\mu_2\) (non-atomicity of a measure means that every measurable set can be divided into two parts of equal measure).
- A linear continuous operator \(A\), acting from \(\mathscr L_p\) into \(\mathscr L_q\), will be called improving if
\[ \lim_{\mu_2(F)\to 0}\|P_F A\|=0. \tag{1} \]
Improving operators are operators that transform norm-bounded sets \(\mathfrak M\) of functions of the space \(\mathscr L_p\) into sets \(A\mathfrak M\) of functions of the space \(\mathscr L_q\) with uniformly absolutely continuous norms. There are no nonzero improving operators acting from \(\mathscr L_p\) into \(\mathscr L_\infty\).
A linear continuous operator \(A\), acting from \(\mathscr L_p\) into \(\mathscr L_q\), will be called co-improving if
\[ \lim_{\mu_1(G)\to 0}\|AP_G\|=0. \tag{2} \]
Co-improving operators are operators that transform every norm-bounded sequence of functions \(x_n\in\mathscr L_p\) converging in measure to \(x_0\) into a sequence of functions \(Ax_n\), strongly converging
to $Ax_0$ in $\mathscr L_q$. There do not exist nonzero co-improving operators acting from $\mathscr L_1$ to $\mathscr L_q$ for $q \geqslant 1$.
If $1 \leqslant p < \infty$, $1 \leqslant q \leqslant \infty$, then an operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ is improving if and only if the adjoint operator $A^*$, acting from $\mathscr L_{q'}$ to $\mathscr L_{p'}$, is co-improving.
We shall say that a continuous operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ has the Ando property if
\[ \lim_{\mu_1(F)+\mu_2(G)\to 0} \|P_G A P_F\|=0 . \tag{3} \]
Improving and co-improving operators have the Ando property.
Theorem 1. Let $1 \leqslant q < p < \infty$. Then every linear continuous operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ has the Ando property.
This assertion for integral operators with nonnegative kernels was proved by T. Ando in [6].
2. A linear operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ is called regular (see [7]) if it can be represented in the form
\[ A = A_1 - A_2, \]
where $A_1$ and $A_2$ are positive linear operators acting from $\mathscr L_p$ to $\mathscr L_q$. Every regular operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ is continuous (see [8]). An integral operator with kernel $K(t,s)$, acting from $\mathscr L_p$ to $\mathscr L_q$, is regular if and only if the integral operator with kernel $|K(t,s)|$ acts from $\mathscr L_p$ to $\mathscr L_q$.
Theorem 2. Let a linear regular operator $A$ act from $\mathscr L_p$ to $\mathscr L_q$, where $0 < p \leqslant \infty$, $0 < q < \infty$, and have the Ando property. Then $A$ is an improving operator.
Theorem 3. Let a linear regular operator $A$ act from $\mathscr L_p$ to $\mathscr L_q$, where $1 < p \leqslant \infty$, $0 < q \leqslant 1$. Then $A$ is a co-improving operator.
In particular, it follows from Theorems 1–3 that every regular operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ for $1 < p < \infty$, $p > q$, is simultaneously improving and co-improving.
Theorem 4. Let $A$ be a regular linear operator acting from $\mathscr L_p$ to $\mathscr L_q$ $(0 < p,q < \infty)$.
Then the operator $A$ maps every set $\mathfrak M$ of functions with equiabsolutely continuous norms in $\mathscr L_p$ into the set $A\mathfrak M$ of functions with equiabsolutely continuous norms in $\mathscr L_q$.
This theorem was proved jointly with E. I. Pustylnik (see also [5]).
3. A linear operator $A$ acting from $\mathscr L_p$ to $\mathscr L_q$ is called compact in measure (see [3]) if it maps every norm-bounded set $\mathfrak M \subset \mathscr L_p$ into a set compact in measure. For compactness in measure of an operator $A$ acting from $\mathscr L_p$, where $p>1$, to $\mathscr L_q$, it is sufficient that $A$ transform every sequence $x_n \in \mathscr L_p$ weakly converging to $x_0$ into a sequence $Ax_n$ converging to $Ax_0$ in measure. For $q \geqslant 1$ the converse assertion is true: every operator acting from $\mathscr L_p$ to $\mathscr L_q$ and compact in measure transforms a sequence $x_n \in \mathscr L_p$ weakly converging to $x_0$ into a sequence $Ax_n$ converging to $Ax_0$ in measure.
Let $A$ be a continuous and compact-in-measure operator acting from $\mathscr L_p$ to $\mathscr L_q$ $(1 \leqslant p,q \leqslant \infty)$. In the general case the adjoint operator $A^*$, acting from $\mathscr L_{q'}$ to $\mathscr L_{p'}$, need not have the compactness-in-measure property. If, however, the operator $A$ is regular and $1 \leqslant p < \infty$, $1 < q < \infty$, then the operator $A$ is compact in measure if and only if the operator $A^*$ is compact in measure. The regularity condition on the operator $A$ can be replaced by the weaker assumption that $A$ maps every set of functions with equiabsolutely continuous norms in $\mathscr L_p$ into a set of functions with equiabsolutely continuous norms in $\mathscr L_q$.
In (3) it is shown that every regular integral operator acting from \(\mathscr L_p\) into \(\mathscr L\) is completely continuous if \(1 < p \leq \infty\). From Theorem 3 it follows that every integral operator acting from \(\mathscr L_p\) into \(\mathscr L_q\), where \(1 < p \leq \infty,\ 0 < q \leq 1\), is completely continuous. From these assertions it follows that every regular integral operator is compact in measure. We do not know whether there exist nonregular integral operators which do not have the property of compactness in measure. Let us note that an integral operator \(A\) is compact in measure if
\[ \lim_{\mu_1(D)\to 0}\ \sup_{\|x\|\leq 1}|AP_Dx|=0, \tag{4} \]
where
\[ |z|=\inf_\alpha\{\alpha+\mu_2[t:\ |z(t)|\geq \alpha]\}. \]
In particular, condition (4) is satisfied if the operator \(A\) can be extended to a continuous operator acting from some \(\mathscr L_{p_1}\), where \(p_1<p\), into some \(\mathscr L_{q_1}\).
The property of complete continuity of linear operators acting from \(\mathscr L_p\) into \(\mathscr L_q\), \(q<\infty\), may be regarded as a combination of the property of compactness in measure and the property of improvability (or, if \(1<p<\infty,\ 1\leq q\leq \infty\), as a combination of the property of compactness in measure of the adjoint operator and the property of co-improvability).
Theorem 5. A linear operator \(A\) acting from \(\mathscr L_p\) into \(\mathscr L_q\) is completely continuous if it has the Ando property and if one of the following conditions is fulfilled:
a) \(A\) is regular and compact in measure, \(q<\infty\);
b) \(A\) is regular, \(A^*\) is compact in measure, \(1<p<\infty,\ 1\leq q\leq \infty\);
c) \(A\) and \(A^*\) are compact in measure, \(1<p,\ q<\infty\).
The complete continuity of a regular integral operator acting from \(\mathscr L_p\) into \(\mathscr L_q\), where \(1<q<p<\infty\), was proved in \((^6)\) by T. Ando (he also considered operators acting in Orlicz spaces).
- Every norm-bounded sequence \(x_n\in\mathscr L_p\) converging in measure converges weakly if \(p>1\). Therefore operators \(A\) acting from \(\mathscr L_p\) into \(\mathscr L_q\) \((1<p<\infty,\ 1\leq q\leq \infty)\) which are compact in measure transform norm-bounded sequences \(x_n\in\mathscr L_p\) converging in measure to \(x_0\) into sequences \(Ax_n\in\mathscr L_q\) which also converge in measure.
Theorem 6. Let \(A\) be an operator compact in measure, acting in \(\mathscr L_p\) \((1<p<\infty)\).
Then the operator \(T=I-A\) transforms every norm-bounded and measure-closed set \(\mathfrak M\subset\mathscr L_p\) into a set \(T\mathfrak M\) closed in measure.
It follows from Theorem 6 that the operator \((I-A)^{-1}\), if it exists, is representable in the form \(I-B\), where \(B\) is an operator compact in measure.
- In applying the theorems of the present article one should keep in mind that there are no nonzero linear continuous operators acting from \(\mathscr L_p\) into \(\mathscr L_q\) if \(p<1,\ p<q\). There do not exist nonzero linear integral operators with nonnegative kernels acting from \(\mathscr L_p\) into \(\mathscr L_q\) if \(p<1\). We do not know whether there exist nonzero completely continuous operators acting from some \(\mathscr L_p\), where \(p<1\), into some \(\mathscr L_q\).
The author expresses his gratitude to M. A. Krasnosel’skii, under whose supervision he is working.
Voronezh State University
Received
6 VI 1964
CITED LITERATURE
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