7 1 0 2 b e F 7 2 STRICT S-NUMBERS OF THE VOLTERRA OPERATOR ] A O¨ZLEMBAKS¸I1,2,TAQSEERKHAN1,3,JANLANG1,ANDVÍTMUSIL1,4 F . h t Abstract. For Volterra operator V: L1(0,1) → C[0,1] and summation opera- a tor σ: ℓ1 → c, we obtain exact values of Approximation, Gelfand, Kolmogorov, m MityaginandIsomorphismnumbers. [ 2 v 1 1. introduction and main results 1 Compactoperatorsandtheirsub-classes(nuclearoperators,Hilbert-Schmidtopera- 1 1tors, etc.) play a crucialrole in many different areas of Mathematics. These operators 0are studied extensively but somehow less attention is devoted to operators which are .non-compactbutcloseto the classofcompactoperators. Inthis workwe willfocuson 1 0two such operators. 7First, consider the Volterra operator V, given by 1 t v: Vf(t)= f(s)ds, (0≤t≤1), for f ∈L1(0,1). (1.1) Z i 0 X When V is regarded as an operator from Lp into Lq, (1 < p,q < ∞), it is a compact aroperator,howeverinthelimitingcase,whenV mapsL1 intothespaceC ofcontinuous functions on the closedunit interval, the operatoris bounded, with the operatornorm kVk = 1, but non-compact. It is worth mentioning that, despite being non-compact or even weakly non-compact, this operator possesses some good properties as being strictlysingular(followsfrom[BG89],orsee[Lef17]). ThismakesVolterraoperator,in the above-mentioned limiting case, an interesting example of a non-compact operator “close” to the class of compact operators. The focus of our paper will be on obtaining exact values of strict s-numbers for this operator. Volterraoperatorwasalreadyextensivelystudied. Letus brieflyrecallthose results relatedto ourwork. The firstcreditgoesto V.I.Levin[Lev38], whocomputed explic- itly the norm of V between two Lp spaces (1 < p < ∞) and described the extremal functionwhichisconnectedwiththefunctionsin . Lateron,E.Schmidtin[Sch40]ex- p tendedthisresultforV : Lp →Lq,where1 0. By the very definition of the n-th Kol- mogorov number, there exists a subspace of C, say N, such that dimN 0befixed. Bythedefinitionofthen-thGelfandnumber, we can find a subspace M in L1 having codimM 0 be fixed. We have kfk−flkL1/M <η (3.12) for k and l sufficiently large. Let us denote f = 1(f −f ) for these k and l. Thanks 2 k l to (3.12) and the definition of quotient norm, one canfind a function g ∈M such that kf −gk ≤η. 1 On setting g h= , 1+η we have 1 khk ≤ kfk +kf −gk ≤1, 1 1 1 1+η (cid:0) (cid:1) STRICT S-NUMBERS OF THE VOLTERRA OPERATOR 9 whence h∈B . Next M 1 kVhk ≥ kVfk −kV(f −g)k ∞ ∞ ∞ 1+η (cid:0) (cid:1) 1 1 ≥ −kVkkf −gk 1 1+η(cid:18)2 (cid:19) 1 1 ≥ −η , 1+η(cid:18)2 (cid:19) thus 1 1 sup kVfk ≥ −η ∞ 1+η(cid:18)2 (cid:19) f∈BM and the lemma follows, since η >0 was arbitrarily chosen. The proof of the discrete counterpart(ii) is completely analogous,once we consider the canonical vectors ek instead of f , hence we omit it. (cid:3) k Lemma 3.5. Let n≥2. Then we have the following upper bounds of the Approxima- tion numbers of V and σ. (i) 1 a (V)≤ ; n 2 (ii) 1 a (σ)≤ . n 2 Proof. (i) Consider the one-dimensional operator F: L1 →C given by 1 1 Ff(t)= f(s)ds, (0≤t≤1), for f ∈L1. 2Z 0 Then F is a sufficient approximationof V. Indeed, t 1 1 kVf −Ffk = sup f(s)ds− f(s)ds ∞ 0≤t≤1(cid:12)(cid:12)Z0 2Z0 (cid:12)(cid:12) (cid:12)(cid:12)1 t 1 1 (cid:12)(cid:12) = sup f(s)ds− f(s)ds 0≤t≤1(cid:12)(cid:12)2Z0 2Zt (cid:12)(cid:12) (cid:12)(cid:12)1 t 1 1 (cid:12)(cid:12) ≤ sup |f(s)|ds+ |f(s)|ds 0≤t≤12Z0 2Zt 1 = kfk 1 2 and therefore 1 a (V)≤kV −Fk≤ . n 2 In order to show (ii), choose the operator ∞ 1 ̺(x) = x , (k ∈N), for x∈ℓ1. k j 2 Xj=1 10 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL This is a well-defined one-dimensional operator and, by the calculations similar to above, kσ−̺k≤1/2. The proof is complete. (cid:3) Now, we are at the position to prove the main results. Proof of Theorem 1.1. Let n ≥ 2 be fixed. Since a (V) is the largest between all n s-numbers,weimmediatelyobtaintheinequalitiesa (V)≥c (V)anda (V)≥d (V). n n n n Using Lemma 3.5 with Lemma 3.4 and Lemma 3.3 we obtain 1 ≥a (V)≥c (V)≥ 1 and 1 ≥a (V)≥d (V)≥ 1 2 n n 2 2 n n 2 respectively. This gives (1.3). Next, let n be arbitrary. Due to i (V) being the smallest strict s-number, we have n thati (V)≤b (V)andalsoi (V)≤m (V). Forthelowerbound,weuseLemma3.1, n n n n while fortheupper, wemakeuse ofLemma3.2andthe resultofLefèvre,[Lef17]. This gives (1.4). (cid:3) Proof of Theorem 1.2. The proof follows along exactly the same lines as that of Theorem 1.1 and hence omitted. (cid:3) References [AHLM16] I. A. Alam, G. Habib, P. Lefèvre, and F. Maalouf. Essential norms of Volterra and Cesàro operators on Mu¨ntz spaces. arXiv:1612.03218,2016. [BG89] J. Bourgain and M. Gromov. Estimates of Bernstein widths of Sobolev spaces. InGeometricaspects offunctional analysis (1987–88), volume1376 of Lecture Notes in Math., pages 176–185.Springer, Berlin, 1989. [BS67] M. Sˇ. Birman and M. Z. Solomjak. Piecewise polynomial approximations of functions of classes Wα. Mat. Sb. (N.S.), 73 (115):331–355,1967. p [CS90] B. Carl and I. Stephani. Entropy, compactness and the approximation of operators, volume 98 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1990. [EE87] D. E. Edmunds and W. D. Evans. Spectral theory and differential operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. [EE04] D. E. Edmunds and W. D. Evans. Hardy operators, function spaces and embeddings. SpringerMonographsinMathematics.Springer-Verlag,Berlin, 2004. [EL06] D. E. Edmunds and J. Lang. Approximation numbers and Kolmogorov widths ofHardy-typeoperatorsina non-homogeneouscase. Math. Nachr., 279(7):727–742,2006. [EL07] D. E. Edmunds and J. Lang. Bernstein widths of Hardy-type operators in a non-homogeneous case. J. Math. Anal. Appl., 325(2):1060–1076,2007. [EL11] D. E. Edmunds and J. Lang. Eigenvalues, embeddings and generalised trigonometric functions, volume 2016 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011.

Strict s-numbers of the Volterra operator

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