数理解析研究所講究録
979 巻 1997 年 34-41

34

ON THE STRUCTURE OF CERTAIN CLASS OF VOLTERRAINTEGRAL OPERATORS AND ESTIMATES OF APPROXIMATIONNUMBERS$\mathrm{V}.\mathrm{D}$. Stepanov1

Computer Center of the Far-Eastem Bmnch of the Russian Academy of Sciences
Khabarovsk, Russia

Introduction.

VVe study the integral operators of the form
$Kf(x)=v(x) \int_{0}^{x}k(X, y)u(y)f(y)dy$,.

(1)$x>0_{l}$weight real functions $v(t)$ and $u(t)$ are locally integrable and the kernel $k(x, y)\geq 0$ satisfies
where
the following condition: there exists a constant $D\geq 1$ such that$\mathrm{t}\dot{\mathrm{h}}\mathrm{e}$$D^{-1}(k(x, y)+k(y, z)\mathrm{I}\leq k(x, z)\leq D(k(x, y)+k(y, Z)),$$x>y>z\geq 0$ ,does not depend on.
A few standardof the kernel $k(x, y)\geq 0$ satisfying (2) are(1) $k(_{X,y})=(X-y)^{\alpha},$,$k(x,y)=1\mathrm{o}_{\mathrm{o}}fl(1+x-y)$(2),,

w.here$D$$x,$ $y,$

(2)

. :

$z$

$\mathrm{e}\mathrm{x}\mathrm{a}\dot{\mathrm{m}}_{\mathrm{P}^{1}\infty}$$\alpha\geq 0$$k(x, y)=10_{\{\supset}^{\sigma} \beta(\frac{x}{y})|\beta\geq 0$(3) $k(x, y)=( \int_{y}^{x_{h(}}s)d_{S})\alpha,$$\alpha\geq 0,$$h(s)\geq 0$,

as well as their various combinations. However, for instance, the kernel of the first kind with negativevalue of does not satisfy (2).
The operators (1) with kernels satisfying (2) were intensively studied during the last decade andmany authors made contributions in this topic, e.g. see the author’s survey
with history andliterature given there, and where themapping properties of (1) were investigated.
Here, in Section 1, we give further extension of some characterization results ofon the Banachfunction spaces for the following problems:$\alpha$$[\mathrm{S}\mathrm{t}_{1}]$$L^{\mathrm{P}}-Lq$$[\mathrm{S}\mathrm{t}_{1}]$(B) Boundedness,(C) Compactness and measure of non-compactness.
In Section 2 we give the more detailing structural results for the operators (1), when they arecompact in the Lebesgue spaces, namely(S) Two-sided estimates of the Schatten-von Neumann ideal norms,(N) Asymptotic behaviour and two-sided estimates ofof approximation numbers.
$\Psi^{\mathrm{n}\mathrm{o}\mathrm{r}}\mathrm{m}\mathrm{s}$1. Problems (B) and (C) in Banach function spaces.
Assume $X$ and be twc Banach spaces of measurable functions defined on. Firstwe consider the problem of the boundedness $K:Xarrow Y$ for the integral operator (1). This case was$Y$$\mathrm{R}^{+}=(0, \infty)$

recently investigated by E. Berezhnoi [research work was$\mathrm{B}\mathrm{e}\mathrm{r}|$, who, in particular, characterized the weak type estimates

supported by INTAS project 94-881.

$\overline{\mathrm{l}\mathrm{T}\mathrm{h}\mathrm{e}}$partiallv

35

for operator (1) with the kernel $k(x, y)\geq 0$ increasing with respect to the first variable and alsosatisfy -condition (see Definition
the strong estimates, when $k(x, y)=1$ and the spaces $X$ andsufficient conditions for thehas also obtained some necessary
3 below). E. Berezhnoi [$k(x,y)\geq0$, stronger then (2).
boundedness of operators (1) with the restrictions on
Definition 1 [BS]. A real normed linear space $X=\{f$ : $||f||x<\infty\}$ of Lebesgue-measurable
is called a Banach function space (BFS), if in addition to the usual norm axioms
functions on
satisfies the following properties:, and $f\in X$ if, and only if,
is defined for every Lebesgue-measurable function on
1)$||f||_{X}<\infty$ ; and $||f||_{X}=0$ if, and only if, $f=0$ almost everywhere
2) $||f||x=|||f|||x$ for all $f\in X_{1}$, then3) ifa-e.,
then $||f_{n}||x\uparrow||f||xi$4) if$mesE<\infty$
;
,
then
5) if
$mesE<\infty$$\int_{E}f(X)dx\leq
C_{E}||f||X$ for all $f\in X$ .
,
then
6) if
$X$ , its associate space $X’$ is defined by
Given BFS
$Y$$\ell$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{B}\mathrm{e}\mathrm{r}|$$\mathrm{R}^{+}$$||f||\mathrm{x}$$\mathrm{R}^{+}$$f$$||f||\mathrm{x}$$(\mathrm{a}.\mathrm{e})|$$0\leq f\leq g\mathrm{a}.\mathrm{e}.$$||f||\mathrm{x}\leq||g||_{Xi}$$0\leq f_{n}\uparrow f$$||\chi_{E}||\mathrm{x}<\infty$$X’= \{g:\int_{0}^{\infty}|fg|<\infty$

for all $f\in X\}$and endowed with the associate norm$||g||x^{J}= \sup\{\int^{\infty}0\leq|fg|:||f||x1\}$.

is also the Banach function space satisfying axioms (1-6) and, moreover, $X’$ is the norm funda, that is the inequality
mental subspace of the dual space$X’$$X^{*}$$||f|| \mathrm{x}--\sup\{\int_{0}\infty f|g|:||g||x’\leq 1\}$.
holds for all.
$X’$ are the complete normed linear spaces and$X,$
spacesThe
for every sequence$X$ has absolutely continuous norm (AC norm), if for all $f\in X,$$X’$ and
have
that
paper
the
assume
.
We
such,
that
of sets.
theLet be a it Banach sequence space (BSS), what means that axioms (1-6) are fulfilled with respectdenote the standard basis in .
to the count measure and let$X$
Definition 2. Given BFS and BSS $X$ is said to be -concave, if for any sequence of disjoint, and for all $f\in X$
suchintervals$f\in X[\mathrm{B}\mathrm{S}]$$X”=X[\mathrm{B}\mathrm{S}]$$||f\chi_{E_{\mathrm{R}}}||xarrow 0$$\{E_{n}\}\subset \mathrm{R}^{+}$$Y$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{o}\mathrm{u}_{\mathrm{o}}\sigma \mathrm{h}_{0}\mathrm{u}\mathrm{t}$$\chi_{E_{n}}arrow 0\mathrm{a}.\mathrm{e}$$\mathrm{A}\mathrm{C}-\mathrm{n}\mathrm{o}7ms$$\ell$$p$$\{e_{n}\}$$l,$$\{J_{k}\}$$\ell$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\cup J_{k}=\mathrm{R}^{+}$$|| \sum_{k}e_{k}||x_{J}kf||_{X}||\ell|\leq d1||f|x$

independent on $f\in X$ andwhere a finite positive constantsuch
be –convex, if for any sequence of disjoint intervals,$\{J_{k}\}$$d_{1}$$\{I_{k}\}$$\ell$$||g||_{Y} \leq d_{2}||\sum_{k}$.

$e\kappa.||\chi tkg||_{Y}||\ell$. Analogously, BFS $Y$ is said to.
, and for all $g\in X$$=\mathrm{R}^{+}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\cup I_{k}$36with a finite positive constant
independent on $g\in Y$ and.
Definition 3 [Ber]. We say, that Banach function spaces $X,$ satisfy –condition, if there exista Banach sequence space such that $X$ is -concave and is -convex simultaneously.
$0/0,$
Throughout the paper the uncertainties of the form
are taken equal to zero,the inequality $A<<B$ means $A\leq cB$ , where depends only on $D$ and, possibly, on
andfrom
Definition however the relationship $A\approx B$ is interpreted as $A<<B<<A$ or $A=cB$ .
denotesthe characteristic function of a set.
$d_{2}$$\{I_{k}\}$

$Y$

$Y$

$p$

$\ell$$\ell$$\ell$$0\cdot\infty,$$\infty/\infty$$c$$d_{1}$$d_{2}$$2_{1}$$\chi_{E}$$E\subset \mathrm{R}^{+}$

1.1. Boundedness. Put for all $t\geq 0$(3)$A_{0}= \sup_{0t>}A0(t)=\sup_{0t>}||x1t,\infty|v||Y||\chi[0,t|(\cdot)k(t, \cdot)u(\cdot)||x’,$$A_{1}= \sup_{t>0}A1(t)=\sup_{>t0}||\chi \mathfrak{l}t,\infty|(\cdot)k(\cdot, t)v(\cdot)||_{Y}||x_{\mathrm{t}0,t}|u||_{x}$,(4)and let $A=A_{0}+A_{1}$ . Note, that $A_{0}=A_{1}$ , if $k(x, y)=1$ .
THEOREM 1.1. Let $X$ and be BFS satisfying the -conditi on and let $K$ be the integral operator
theof form (1) with the kemel $k(x, y)\geq 0$ satisfying (2). Then $K$ : $Xarrow Y$ is bounded, if and only if,
is finite. $iVloreover$,
$Y$

$l$

$A$

$||K||_{Xarrow Y}\approx A$.

(5)$Y=L^{q},$ $1\leq p,$
Example 1. If
are thespaces with the usual norms,
then the -condition holds if, and only if, $p\leq q$ and quantities (3) and (4) transform into
$X=L^{\mathrm{p}},$$q\leq\infty$$\mathrm{L}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{s}_{\mathrm{t}\supset}\sigma \mathrm{u}\mathrm{e}$$\ell$$A_{0}= \sup_{0t>}A\mathrm{o}(t)=\sup_{t>0}(\int_{t}^{\infty}k^{q}(_{X}, t)|v(x)|qdx)^{1/p’}q(\int^{t}\mathrm{o}u|(y)|p’dy)^{1}/$$A_{1}= \sup_{t>0}A_{1}(t)=\sup_{t>0}(\int_{t}^{\infty}|v(x)|\sigma dx\mathrm{I}^{1/q}(\int_{0}^{t}k^{p}(t, y)|u(y)|py\prime\prime d)1/p’$in this case, where$\underline{1}+\underline{1}=1,$$p$criterion is true$p’$$\underline{1}+\underline{1}=1$

$q$

$q’$

$[\mathrm{s}_{\mathrm{t}_{1}}]:||K||_{Larrow}PL^{\mathrm{q}}\approx B$. For the case $1<q<p$ in$L^{p}$–$L^{q}$

setting the$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\sigma 0$, where $B=B_{0}+B_{1}$ defined by$B_{0}=\{$ $\int_{0}\infty(\int_{t}\infty\int_{0}k^{q}(_{X}, t)|v(x)|^{q}dX)^{\Gamma/}q(td|u(y)|^{p’}y)r/q’u|(t)|\mathrm{p}J\}dt1/\Gamma$$B_{1}=\{$ $\int_{0}^{\infty}(\int_{t}^{\infty}|v(x)|q)^{r}d_{\mathcal{I}}(/\mathrm{P}\int 0)p’|kp’(t, y)|u(y)|dyvt\}r/_{\mathrm{P}’}(t)|qtd1/r$

where$\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$$0<q<p,$. If $k(x, y)=1$ , then$p\geq 1[\mathrm{S}]$$B_{0}=(_{q}\mathrm{E}_{-}^{J})^{1/r}B_{1}$with suitable modification, when

and the above criterion is valid for the range$p=1[\mathrm{S}\mathrm{S}]$.

$\mathrm{L}\circ \mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}_{\mathrm{Z}\mathrm{s}}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}LrSL_{1}\equiv s(\varphi\rho)r\mathrm{R}+\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{a}1\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{a}\mathrm{b}[\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\circ \mathrm{n}\mathrm{S}f\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}||f|\mathrm{c}\circ \mathrm{n}\mathrm{s}|_{\Gamma s}\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{m}\mathrm{p}1\mathrm{e}2\mathrm{F}_{0}\mathrm{r}0<r<\infty,0<s_{1}\leq\infty \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}1\mathrm{o}\mathrm{c}\mathrm{a}11\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{b}1\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\varphi,(x)\varphi$

on$<\infty$$||f||_{r\delta,\varphi}=( \int_{0}\infty f(t^{1/\prime}..(t)\mathrm{I}^{s}\frac{dt}{t})1/s$

for $0<s<\infty$ ,, the, where$\mathrm{R}^{+}$

37$||f||_{\Gamma s}, \varphi\sup_{\mathrm{f}>^{0}}t=/rf1\cdot\cdot(t)$for$s=\infty$,and$f \cdot\vee(t)=\frac{1}{t}\int_{0}^{t}f^{\mathrm{r}}(S)ds$$f^{*}(t)= \inf\{x>0$ :If $r=s,$,.

$\lambda_{f}(x)=\int_{\{r:}\tau\in \mathrm{R}^{+}|f(y)|>x\}\varphi(Z)dz\leq t\}$the.n$||f||_{\Gamma\gamma\varphi},=( \int_{0}^{\infty}|f(X)|^{r}\varphi(_{X})d_{X})^{\iota}/r$

then the -condition holds if, and only if, $\max(r, s)\leq\min(p, q)$ and in this case
If
$K$
$Xarrow
Y$ is sandwiched by $A=A_{0}+A_{1}$ , where
the norm of :$\ell$$X=L_{\varphi}^{rs},$ $Y=L_{\psi}^{pq}$$A_{0=\sup_{t>0}}A_{0(t})=||x_{[0,t\mathrm{I}}(\cdot)k(t, \cdot)(u(\cdot)/\varphi(\cdot))||_{r’s’},\varphi||\chi_{[}t,\infty \mathrm{i}^{v||_{p}}q,\psi$$A_{1}= \sup_{0t>}A1(t)=||\chi[0,t|(u/\varphi)||_{r^{J}s^{J}\varphi}||x[t,\infty|(\cdot)k(\cdot, t)v(\cdot)1||pq,\psi$’.

Remark 1.1. (i) If the -condition fails, then the lower bound in (5) is nevertheless true. However,there exist an operator, when (5) is valid for the spaces with no -condition. Indeed, if we take $k(x, y)=$$v(x)=1$ , then in Lorentz space setting above the criterion (5) holds for $1<r=s,$ $q\geq r,$ $0<p<\infty$([Sa], Theorem 2).
(ii) If $1<r=s,$ $0<q<r<\infty,$ $0<p<\infty$ , then ’the criterion for the boundedness of this
, Theorem 2.2). Put $Uf(x)= \int_{0}^{x}f(y)u(y)dy$ . Then
operator is the following ($\ell$$p$$1,$$[\mathrm{S}\mathrm{t}_{1}]$$||Uf||_{\mathrm{P}q,\psi}$$( \int_{0}^{\infty}(||\chi[t,\infty|||pq,\psi)\gamma(d||x_{[}\mathrm{o},t|(u/\varphi)||\Gamma J)^{\gamma}r’,\varphi)1/\gamma$$\sup_{f_{f}^{A}0rr,\varphi}\approx\overline{||f||}$where$\frac{1}{\gamma}=\frac{1}{q}-\frac{1}{r}$.

1.2. Compactness and measure of non-compactness.
THEOREM 1.2. Let the assumptions of Theorem 1 be fulfilled and the spaces $X’$ andthe $AC$-norms. Then the operator $K:Xarrow Y$ is compact if, and only$\lim_{tarrow a}.A_{i}(t)=\lim_{tarrow b}.A_{i}(t)=0$

where$a_{i}= \inf\{t>0:A_{i}(t)>0\}$ ,,$\dot{i}=0,1$$\dot{i}f,$

$A$

$Y$

have

is finite and(6),$b_{i}= \sup\{t>0:A_{i}(t)>0\}i=0,1$ .

.
Remark 1.1. (i) In fact, it follows from the proof of Theorem 1.2, that $a_{0}=a_{1},$end-points,
it is easy
(6)
thehowever
for
used
to
be
formulatedtheconditionmany
authors(ii) By$b_{0}=b_{1}=\infty$$a_{0}=a_{1}=0,$
,
with
valid
counterexamle,
(6)
is
is
when
finiteandto point out a formal
$K$
of
for
the
end-points
toformulated
is,
(6)
that
the
conditionhas
is non-compact. The matterbutinterval of non-zero action of $K$ .
the(iii) For the case $1<q<p$ in $L^{p}-L^{q}$ setting the operator $K:L^{p}arrow L^{q}$ is compact if, and only
$B<\infty$.
(see Example 1). It follows
the Ando theorem
if,$b_{0}=b_{1}$$A$$\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}|$$\mathrm{f}\mathrm{r}\mathrm{o}\ln$$[\mathrm{A}|$38

In the non-compact case we estimate the measure of non-compactness of $K$ or, equivalently, thedistance between $K$ and the set of finite rank operators defined by$\alpha(K)=\inf||K-P||$ ,where the infimum is taken over all bounded linear maps $P:Xarrow Y$ of finite rank. To this end weneed additional portion of notations. For $0<a<z<b<\infty$ we put$]_{L()=}^{0}a0< \iota<\sup_{a}||x1t,a\mathrm{I}^{v}||Y||\chi[0,t\mathrm{I}(\cdot)k(t, \cdot)u(\cdot)||X’$,$J_{L()||xt,a}^{1}a= \sup_{0<t<a}1|(\cdot)k(\cdot, t)v(\cdot)||_{Y}||\chi_{1^{0}t},|u||_{X}’$,$f_{L}(z)= \max(J_{L(}\mathrm{O}z),$$JL(1)a),$ $JL=, \lim_{arrow a\mathrm{o}}J_{L(z})\sim$ ’,$J_{R}^{\mathrm{o}_{()}}b= \sup_{<b<t\infty}||\chi_{|a|}t,v||_{Y}||x\{0,t|(\cdot)k(t, \cdot)u(\cdot)||_{X}’$$J_{R}^{1}(b)= \sup_{<bt<\infty}||\chi_{[t,a}\mathrm{I}(\cdot)k(\cdot, t)v(\cdot)||_{Y}||\chi_{||}0,tu||_{X}’$,$J_{R}(Z)= \max(J_{R}^{0_{(z}1}),$$JR(a)),$ $JR= \lim_{zarrow a_{0}}J_{R}(Z)$,$J= \max(J_{L}, J_{R})$.

THEOREM 1.3. Let the assumptions of Theorem 2 be valid and $K:Xarrow Y$ be bounded. Then.

$D^{-1}J\leq\alpha(K)\leq(d_{1}, d_{2}, D)J$

2. Problems (S) and (N) in Lebesgue spaces

2.1. Schatten-von Neumann ideal norms. Let $H$ be a separable Hilbert space. Then theset of all linear bounded operators $T:Harrow H$ forms the normed, where-the ideal of
all compact operators. The theory of simmetrically normed
idealswas developed by. functions defined on the space of sequences with a finite number of non-zero terms
using the$\mathrm{a}1_{\mathrm{o}}\propto \mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{B}$$(\mathrm{s}.\mathrm{n}.)$$\Phi$$\mathrm{s}.\mathrm{n}$([GK], Chapter 3). Ifof$T\in\sigma_{\infty}$, then$\tau*\in\sigma_{\infty}$

and$(\tau^{*}T)^{1}/2\in\sigma_{\infty}$

numbers $s_{j}(T)=\lambda_{j}[(T^{\cdot}T)1/2]$ were used, with the$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$\sigma_{\infty}$$\sigma_{\Phi}\subset\sigma_{\infty}$. To construct$\sigma_{\Phi}$$\mathrm{e}\mathrm{i}_{1\supset}\sigma \mathrm{e}\mathrm{n}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{a}\mathrm{e}/\backslash _{j}\geq 0$

the sequences

taken according

to their multiplicity and decrease. Formula $||T||_{\sigma 0}=\Phi(s_{j}(T))$ defines the norm (quasinorm) in the. ideal . The most well-known are the . ideals
related to the space of sequences ,$\mathrm{s}.\mathrm{n}$$\sigma_{\Phi}$$0<p\leq\infty$$\mathrm{s}.\mathrm{n}$. The norm (quasinorm)

Neumann norm (quasinorm). Thus,$l_{p}$$\sigma_{p}$$||T||_{\sigma_{\mathrm{P}}}=( \sum_{j}s_{j}p(T))^{1/\mathrm{P}}$$||T||_{\sigma \mathrm{z}}=||T||$

and$||T||_{\sigma_{2}}$is usually called by the Schatten-vonis the Hilbert-Schmidt norm expressed

for an integral operator $Tf(x)= \int T(x, y)f(y)dy$ by the formulae$||T||_{\sigma \mathrm{z}}=( \int\int|\tau(X, y)|^{2}dXdy)^{1/2}$of an integral operator substansially depends on
It is known [BS], that in general the norm
the smoothness of its kernel, when $p<2$ , however for some particular operators of complex harmonic
estimates of the Schatten-von Neumann norms are well known,
analysis the effective.
$||T||_{\sigma,}$$\mathrm{t}\mathrm{w}\mathrm{c}\succ \mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{d}$$\mathrm{e}.\mathrm{g}$

39

and
aim of the section is to present a brief account of some results from
(2) forcondition
the
(1)
with
operators
integralabout the Schatten-von Neumann ideal norms for thetheir kernels.
Let $H=L^{2}(0, \infty)$ andsee [Pa],$[\mathrm{S}\mathrm{t}_{2}]$$[\mathrm{E}\mathrm{S}_{2}]$$[\mathrm{P}].\mathrm{T}\mathrm{h}\mathrm{e}$$A_{0}^{2}= \sup_{t>0}\int_{t}^{\infty}k^{2}(X, t)|v(x)|^{2}dx\int_{0}^{t}|u(y)|^{2}dy$,$A_{1}^{2}= \sup_{t>0}\int_{t}^{\infty}|v(x)|^{2}dx\int_{0}^{t}k^{2}(t, y)|u(y)|^{2}dy$.

Theorem 1.1 and the Hilbert-Schmidt formula$\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$||K||_{\sigma_{\infty}}\approx A_{0}+A_{1}$,$||K||_{\sigma_{2}}=( \int_{0}^{\infty}|v(_{X})|^{2}dX\int_{0}x_{k^{2}(t,y)|u(y)|dy})^{1}2/2=(\int_{0}^{\infty}|u(y)|^{2}dy\int_{y}^{\infty}k^{2}(X, y)|v(\mathcal{I})|^{2}d\mathcal{I}\mathrm{I}^{1}/2$

Using these formulas and applying the real method of interpolation we obtain the
THEOREM 2.1. Let $K$ be an operator of the form (1) with the kemel satisfying (2) andThen$\mathrm{f}\mathrm{o}\mathrm{l}1_{0}\mathrm{w}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$K\in\sigma_{\infty}$.

$||K||_{\sigma_{\mathrm{p}}} \approx(\int_{0}^{\infty}[(\int_{0}x_{k^{2}(x,y)|u(y)|dy)^{\mathrm{P}}}/2(2\int x\frac{\mathrm{p}}{2}\infty-1||v(y)|^{2}dy\mathrm{I}v(x)|^{2}+$$( \int_{x}^{\infty}k^{2}(y, X)|v(y)|2dy)^{\mathrm{p}/2}(\int_{0}x8-1||u(y)|2dy)|u(x)2]d\mathcal{I}\mathrm{I}^{1}/p$$2\leq p<\infty$.

(7)

Remark 2.1. The upper bound of (7) is proved in [ES2] and the lower one in [St2]. In casethe formula (7) can be simplified and extended as follows. If$k(x, y)\equiv 1$(8)$Hf(x)=v(X) \int_{0}^{x}f(y)u(y)dy$ ,then$||H||_{\sigma_{\mathrm{p}}} \approx(\int_{0}^{\infty}(\int 0||u(y)dy)2\mathrm{p}/2(\int_{x}^{\infty}x-1)|v(y)|2dy)^{52}|v(x|d_{X}\mathrm{I}1/p$$1<p<\infty$ .

(9)Remark 2.1. In alternate form the equivalence (9) for the case $u(y)=1$ has been established in[N] and later this result has been widely extended in [NS] for the operator$I_{l\text{ノ}}f(x)= \frac{v(x)}{x^{\nu}}\int_{0}^{x}(_{X}-y)\nu-1f(y)dy$of,$\nu>1/2$.

2.2. Approximation numbers. Recall that if $T:Xarrow Y$ , then the n-th approximation number
is defined by$a_{n}= \inf$ $||T-P||$ ,rankP $<n$ }, $n=1,2,${$T$$\ldots$

The problem of asympthotic behaviour of the approximation numbers is well known and was,[ and others and for the operator (8) in the papers
treated in the monograghs [$\mathrm{E}\mathrm{E}|,$$\mathrm{K}|$$[\mathrm{E}\mathrm{E}\mathrm{H}_{1}],$$[\mathrm{E}\mathrm{E}\mathrm{H}_{2}]$40$[\mathrm{E}\mathrm{S}_{1}],$$[\mathrm{L}\mathrm{S}_{1}],$(9).

$[\mathrm{L}\mathrm{S}_{2}]$. Here we present the new result for the operator (8) easy comparable with formula

THEOREM 2.2 Let $1<p,$ $s<\infty$ and the integral operator $H$ :is the sequence of the appronimation numbers of H. Then

$L^{p}(0, \infty)arrow L^{p}(0, \infty)$(8) be compact andgiven by$\{a_{n}\}$$( \sum_{n=1}a^{S})^{1/}nS\approx(\int_{0}^{\infty x}(\int_{0}|u(y)|^{p}dy)\prime S/p^{\prime x}(\int_{0}|v(y)|^{P}dy)\overline{\mathrm{p}}’-1|U(x)|dx)1/s$,$p\neq 2$$\varlimsup_{narrow\infty}no_{n}\leq\gamma_{p}\int_{0}^{\infty}|uv|\leq 2\varliminf_{narrow\infty}na_{n}$$\lim_{narrow\infty}na_{n}=\frac{1}{\pi}\int_{0}^{\infty}|uv|,$

The last two formulas are proved in$[\mathrm{E}\mathrm{E}\mathrm{H}_{2}]$$[\mathrm{E}\mathrm{E}\mathrm{H}_{2}]$.

$p=2$ .

and the proof of the first is based on the results ofRemark 2.2. All the assertions of the paper have naturaland for the dual operator
as well (seefor details).

$\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{o}_{\mathrm{o}}\sigma_{\mathrm{S}}$$(0, \infty)$$K^{*}$,

for a finite interval instead of$[\mathrm{S}\mathrm{t}_{1}]$Acknowledgement. The author wishes to thank Professor Takashi Yoshino for his kind invitation
to give a talk on the workshop ”Recent topics on the operator theory about the structure of operators”
held in the Research Institute of Mathematical Science of Kyoto University, and his warm hospitality
and the travel support.

References[A] T. Ando, On compactness ofoperators, Indag. Math., 24 (1962) 235-239.
[BS] C. Bennett and R. Sharpley, Interpolation of opemtors, Pure Appl. Math. 129, Academic Press,$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\Leftrightarrow}\sigma \mathrm{r}\mathrm{a}1$1988.

[Ber] E.I. Berezhnoi, Sharp estimates of operators on the cones of ideal spaces, Proc. Steklov Inst.
Math. (1994), Issue 3, 3-34.
[BiS] M. Sh. Birman and M.Z. Solomjak, Estimates for thenumbers ofoperators,
Russian Math. Surweys, 32 (1977).
[EE] D.E. Edmunds and W.D. Evans, Spectral theory and differential operators, Oxford: Univ. Press,$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$1987.

$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\circ}\sigma \mathrm{r}\mathrm{a}1$. Edmunds, W.D. Evans and D.J. Harris, Approximation numbers of certain Volterra
integral operators, J. London Math. Soc., (2) 37 (1988), 471-489.
. Edmunds, W.D. Evans and D.J. Harris, Two-sided estimates of the approximati on num[bers of certain Volterm integml opera tors, Centre for mathematical analysis and its applications,
University of Sussex, Research Report No: 96/09, 1996, 25 pp.
. Edmunds and V.D. Stepanov, The measure of non-compactness and approximation numbers of certain Volterra integral operators. Math. Ann. 298 (1994) 41-66.
. Edmunds and V.D. Stepanov, On singular numbers of certain Volterra integral operators.
J. Funct. Anal. 134 (1995) 222-246.
[GK] I.C. Gohberg and M.G. Krein, Introduction to the $theo\eta$linear non-selfadjoint operators,
AMS ’Ranl. Math. Monographs, 18, Providence, R. I., 1969.
[K] H. K\"onig, Eigenvalue distribution of compact opemtors, Birkh\"auser, Boston, 1986.
$[\mathrm{E}\mathrm{E}\mathrm{H}_{1}]\mathrm{D}.\mathrm{E}$$\mathrm{E}\mathrm{E}\mathrm{H}_{2}1\mathrm{D}.\mathrm{E}$$[\mathrm{E}\mathrm{S}_{1}]\mathrm{D}.\mathrm{E}$$[\mathrm{E}\mathrm{S}_{2}]\mathrm{D}.\mathrm{E}$

0.f

41

. Lomakina and V.D. Stepanov, On the compactness and approximation numbers of Hardy. Soc. (2) 53 (1996) 369-382.
type integral operators in Lorentz spaces, J. London
integml opemtors in Banach function
type
Hardy
, On the. Lomakina and V.D.
No: 9-1996, Khabarovsk, 1996,
ReportCenter,Researchspaces on semiaxis, (in russian), Computer
35 pp.
[NS] J. Newman and M. Solomyak, Two-sided estimates on singular numbers for a class of integral
operators on the semi-axis, Integr. Equat. Oper. Th., 20 (1994) 335-349.
Hardy operator, Proc. Amer. Math. Soc.,[N] K. Nowak, Schatten ideal behaviour of a
118 (1993) 479-483.
$[\mathrm{L}\mathrm{S}_{1}]\mathrm{E}.\mathrm{N}$$\mathit{1}\nu \mathit{1}ath$$\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{o}\downarrow’$$[\mathrm{L}\mathrm{S}_{2}]\mathrm{E}.\mathrm{N}$$\mathrm{t}\supset\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{Z}\mathrm{e}\mathrm{d}$operators with holomorphic kernels, (in[Pa] 0. Parfenov, Estimates of singular numbers ofrussian), Bull. St.-Petersburg State Univ., Ser. 1, 8 (1992) 24-32.
0, an investigation of rate offor p[P] V.V. Peller, A discription of Hankel operators of class(1985) 465-494.
Sbomik,
122
Math.
USSR
applications,
other
and
arrpoximation,
rational
Hardy operator, Trans.
fortheinequalities
norm
andLorentz
Lebesgue[Sa] E. Sawyer, Weighted(1984)
281
329-337.
Amer. Math. Soc.
[S] G. Sinnamon, Weighted Hardy and Opial type inequalities, J. lVlath. Anal. Appl., 160 (1991)$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}1$$S_{p}$$>$

433-445.

[SS] G. Sinnamon and V. Stepanov, The weighted Hardy inequality: New proofs and the case p $=1’$.
J. London Math. Soc., (2) 54 (1996) 89-101.
V. Stepanov, Weighted norm inequalities for integml operators and related topics, Nonlinear
1994, 139-176.
analysis, function spaces and applications, vol. 5, Olimpia Press,of certain Volterra
norms
V. Stepanov, On the lower bounds for Schatten-von Neumann
operators, J. London Math. Soc., (to appear).
$[\mathrm{S}\mathrm{t}_{1}]$$\mathrm{P}\mathrm{r}\mathrm{a}_{\mathrm{t}}\sigma \mathrm{u}\mathrm{e}\supset$’$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{e}}\sigma \mathrm{r}\mathrm{a}1$$[\mathrm{S}\mathrm{t}_{2}]$Vladimir Stepanov
Computer Center
Far-Eastern Branch
Russian Academy of Sciences
Shelest 118-205,
Khabarovsk 680042, Russia-mail: 1600@as.khabarovsk.su$\mathrm{E}$