Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 894527 | 11 pages | https://doi.org/10.1155/2012/894527

On Some Applications of a Special Integrodifferential Operators

Academic Editor: Nicolae Popa
Received11 Jan 2010
Accepted19 Oct 2010
Published15 Jan 2012

Abstract

Let 𝐢(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(π‘₯,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} is the unit disc in the complex plane β„‚, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐢𝐴(𝑛)=𝐢𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐢(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐢𝐴(𝑛)=𝐢(𝑛)(𝔻×𝔻)βˆ©β„‹π‘œπ‘™(𝔻×𝔻)). The double integration operator is defined in 𝐢𝐴(𝑛) by the formula βˆ«π‘Šπ‘“(𝑧,𝑀)=𝑧0βˆ«π‘€0𝑓(𝑒,𝑣)𝑑𝑣𝑑𝑒. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator π‘Šβˆ£πΈπ‘§π‘€, where 𝐸𝑧𝑀={π‘“βˆˆπΆπ΄(𝑛)βˆΆπ‘“(𝑧,𝑀)=𝑓(𝑧𝑀)} is an invariant subspace of π‘Š, and study its properties. We also study invertibility of the elements in 𝐢𝐴(𝑛) with respect to the Duhamel product.

1. Introduction and Backgrounds

Let β„‹π‘œπ‘™(𝔻×𝔻) denote the FrΓ©chet space of functions 𝑓(𝑧,𝑀) that are holomorphic in the bidisc 𝔻×𝔻={(𝑧,𝑀)βˆˆβ„‚Γ—β„‚βˆΆ|𝑧|<1and|𝑀|<1}. The product we define on this space is πœ•(π‘“βŠ›π‘”)(𝑧,𝑀)∢=2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0𝑓(π‘§βˆ’π‘’,π‘€βˆ’π‘£)𝑔(𝑒,𝑣)𝑑𝑣𝑑𝑒,(1.1) which obviously defines an integrodifferential operator π’Ÿπ‘“,π’Ÿπ‘“π‘”βˆΆ=π‘“βŠ›π‘”. This product is a natural extension of the Duhamel product on β„‹π‘œπ‘™(𝔻) [1]: 𝑑(π‘“βŠ›π‘”)(𝑧)∢=ξ€œπ‘‘π‘§z0ξ€œπ‘“(π‘§βˆ’π‘‘)𝑔(𝑑)𝑑𝑑=z0π‘“ξ…ž(π‘§βˆ’π‘‘)𝑔(𝑑)𝑑𝑑+𝑓(0)𝑔(𝑧),(1.2) where the integrals are taken over the segment joining the points 0 and 𝑧.

Note that the Duhamel product is widely applied in various questions of analysis, for example, in the theory of differential equations and in solution of boundary value problems of mathematical physics. Wigley [2] showed that, for 𝑝β‰₯1, the Hardy space 𝐻𝑝(𝔻) (which is the space of all holomorphic functions on the open unit disc 𝔻 for which the norm ‖𝑓‖𝐻𝑝=sup0<π‘Ÿ<1ξ‚΅1ξ€œ2πœ‹02πœ‹||π‘“ξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||π‘ξ‚Άπ‘‘πœƒ1/𝑝(1.3) is finite) is a Banach algebra under the Duhamel product βŠ›.

The Hardy space of the polydisc, 𝐻𝑝(𝔻𝑛), is defined as those functions analytic on π”»π‘›βˆΆ=𝔻×⋯×𝔻 for which the following norm is finite: β€–π‘“β€–π»π‘βˆΆ=supπ‘Ÿ1<1β‹―supπ‘Ÿπ‘›<1ξ‚΅1(2πœ‹)π‘›ξ€œ02πœ‹β‹―ξ€œ02πœ‹||π‘“ξ€·π‘Ÿ1π‘’π‘–πœƒ1,…,π‘Ÿπ‘›π‘’π‘–πœƒπ‘›ξ€Έ||π‘π‘‘πœƒ1β‹―π‘‘πœƒπ‘›ξ‚Ά1/𝑝.(1.4) If 𝑝β‰₯1, this is a Banach space, and if 0<π‘Ÿ<1, this is a FrΓ©chet space [3]. In [3], Merryfield and Watson proved that for 𝑝β‰₯1𝐻𝑝(𝔻𝑛) is a Banach algebra with respect to the product (1.1).

In the present paper we prove that the space 𝐸𝑧𝑀 can be given a Banach algebra structure under the Duhamel product (1.1); in particular, we describe the maximal ideal space of the Banach algebra (𝐸𝑧𝑀,βŠ›), where πΈπ‘§π‘€ξ‚†βˆΆ=π‘“βˆˆπΆπ΄(𝑛)ξ‚‡πœ•βˆΆπ‘“(𝑧,𝑀)=𝑓(𝑧𝑀),(1.5)(π‘“βŠ›π‘”)(𝑧𝑀)=2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0𝑓((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑔(𝑒𝑣)𝑑𝑣𝑑𝑒.(1.6) By using product (1.6) we also describe commutant of the operator π‘Šπ‘§π‘€βˆΆ=π‘Šβˆ£πΈπ‘§π‘€, that is, the set of bounded linear operators on 𝐸𝑧𝑀 commuting with π‘Šπ‘§π‘€. Moreover, we describe the set of cyclic vectors of the double integration operator π‘Šπ‘§π‘€ acting on the closed subspace π»π‘π‘§π‘€ξ€½βˆΆ=𝑓(𝑧,𝑀)βˆˆπ»π‘ξ€·π”»2ξ€Έξ€Ύ.βˆΆπ‘“(𝑧,𝑀)=𝑓(𝑧𝑀)(1.7) We recall that a vector π‘₯βˆˆπ‘‹ is called cyclic vector for the operator π΄βˆˆβ„’(𝑋) (Banach algebra of all bounded linear operators on a Banach space 𝑋) if ξ€½spanπ‘₯,𝐴π‘₯,𝐴2ξ€Ύπ‘₯,…=𝑋,(1.8) where span{π‘₯,𝐴π‘₯,𝐴2π‘₯,…} denotes the closure of the linear hull of the set {π‘₯,𝐴π‘₯,𝐴2π‘₯,…}.

2. Description of {π‘Šπ‘§π‘€}ξ…ž

For any operator π΄βˆˆβ„’(𝑋) its commutant {𝐴}ξ…ž is defined by {𝐴}ξ…žβˆΆ={π΅βˆˆβ„’(𝑋)∢𝐡𝐴=𝐴𝐡}.(2.1)

The study of commutant of the concrete operator π΄βˆˆβ„’(𝑋) is one of the important, but generally, not easy problem of operator theory. For this, it is enough to remember the famous Lomonosov’s theorem on the existence of nontrivial hyperinvariant subspace of compact operator 𝒦 on a Banach space 𝑋 (recall that a closed subspace πΈβŠ‚π‘‹ is called hyperinvariant subspace for the operator π΄βˆˆβ„’(𝑋), if it is invariant for any operator 𝐡∈{𝐴}ξ…ž). Note that many papers are devoted to the evident description of commutant (and, more generally, the set of so-called extended eigenvectors [4–6]) for some special operator classes (see, e.g., [7–14]). In this section we describe in terms of the Duhamel operators the commutant of the operator π‘Šπ‘§π‘€ on the closed subspace 𝐸𝑧𝑀 of the space 𝐢𝐴(𝑛). First, we prove the following lemma, which shows that 𝐸𝑧𝑀 is a Banach algebra under the Duhamel product βŠ› given by formula (1.6).

Lemma 2.1. (𝐸𝑧𝑀,βŠ›) is a Banach algebra.

Proof. Indeed, let 𝑓,π‘”βˆˆπΈπ‘§π‘€ be two functions. The norm in 𝐸𝑧𝑀 is defined by β€–π‘“β€–π‘›ξƒ―βˆΆ=maxmax(𝑧,𝑀)βˆˆπ”»2||||πœ•|𝛼|𝑓(𝑧𝑀)πœ•π‘§π›Ό1πœ•π‘€π›Ό2||||∢|𝛼|=𝛼1+𝛼2ξƒ°.=0,1,…,𝑛(2.2) Using (1.6), (2.2) and the Leibnitz formula for the partial derivatives of the product π‘“βŠ›π‘”, it can be proved (which is omitted) that (see e.g., the method of the paper [15, 16]) β€–π‘“βŠ›π‘”β€–π‘›β‰€πΆπ‘›β€–π‘“β€–π‘›β€–π‘”β€–π‘›(2.3) for some constant 𝐢𝑛>0, which proves the lemma.

The main result of this section is the following theorem.

Theorem 2.2. Let π‘‡βˆˆβ„’(𝐸𝑧𝑀) be an operator. Then π‘‡βˆˆ{π‘Šπ‘§π‘€}ξ…ž if and only if there exists a function πœ‘βˆˆπΈπ‘§π‘€ such that 𝑇=π’Ÿπœ‘, where π’Ÿπœ‘ is the Duhamel operator defined by π’Ÿπœ‘πœ•π‘“(𝑧𝑀)=(πœ‘βŠ›π‘“)(𝑧𝑀)=2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0πœ‘((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒.(2.4)

Proof. Let π‘‡βˆˆ{π‘Šπ‘§π‘€}ξ…ž, that is, π‘‡π‘Šπ‘§π‘€=π‘Šπ‘§π‘€π‘‡(2.5) Then we have that π‘‡π‘Šπ‘§π‘€(𝑧𝑀)π‘˜=π‘Šπ‘§π‘€π‘‡(𝑧𝑀)π‘˜,(2.6) for all π‘˜=0,1,…, whence by computing π‘Šπ‘§π‘€(𝑧𝑀)π‘˜ we have π‘‡ξ‚΅ξ€œπ‘§0ξ€œπ‘€0(𝑒𝑣)π‘˜ξ‚Άξ‚΅ξ€œπ‘‘π‘£π‘‘π‘’=𝑇𝑧0π‘’π‘˜ξ‚΅ξ€œπ‘€0π‘£π‘˜ξ‚Άξ‚Άξ€œπ‘‘π‘£π‘‘π‘’=𝑇𝑧0π‘’π‘˜π‘€π‘˜+1ξ‚΅π‘§π‘˜+1𝑑𝑒=π‘‡π‘˜+1π‘€π‘˜+1(π‘˜+1)2ξ‚Ά=1(π‘˜+1)2𝑇(𝑧𝑀)π‘˜+1,(2.7) or 𝑇(𝑧𝑀)π‘˜+1=(π‘˜+1)2π‘Šπ‘§π‘€π‘‡(𝑧𝑀)π‘˜,(2.8) for all π‘˜=0,1,….
From (2.8) we have by induction that 𝑇(𝑧𝑀)π‘˜=π‘Šπ‘˜π‘§π‘€π‘‡1π‘˜ξ‘π‘ =1𝑠2(π‘˜=1,2,…).(2.9) Indeed, for π‘˜=1 we have from (2.8) 𝑇(𝑧𝑀)=π‘Šπ‘§π‘€π‘‡1, as desired.
Assume for π‘˜=𝑛 that 𝑇(𝑧𝑀)𝑛=π‘Šπ‘›π‘§π‘€π‘‡1𝑛𝑠=1𝑠2.(2.10) For π‘˜=𝑛+1 we have from (2.8) that 𝑇(𝑧𝑀)𝑛+1=(𝑛+1)2π‘Šπ‘§π‘€π‘‡(𝑧𝑀)𝑛.(2.11) Now, by considering (2.10) from the latter equality we have 𝑇(𝑧𝑀)𝑛+1=(𝑛+1)2π‘Šπ‘§π‘€ξƒ©π‘Šπ‘›π‘§π‘€π‘‡1𝑛𝑠=1𝑠2ξƒͺ=π‘Šπ‘›+1𝑧𝑀𝑇1(𝑛+1)2𝑛𝑠=1𝑠2=π‘Šπ‘›+1𝑧𝑀𝑇1𝑛+1𝑠=1𝑠2,(2.12) which proves (2.9).
Now, let us show that ξ€·π‘Šπ‘˜π‘§π‘€π‘“ξ€Έξ€œ(𝑧𝑀)=𝑧0ξ€œπ‘€0[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜βˆ’1[](π‘˜βˆ’1)!2𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒.(2.13) For this purpose, first show that ξ€·π‘Šπ‘˜π‘§π‘€π‘“ξ€Έ(𝑧𝑀)=(𝑧𝑀)π‘˜[]π‘˜!2βŠ›π‘“(𝑧𝑀),(2.14) for all π‘˜β‰₯0. Indeed, it follows directly from (1.6) that 1 is the unit with respect to the Duhamel product βŠ› in 𝐸𝑧𝑀, and π‘Šπ‘§π‘€π‘“=π‘§π‘€βŠ›π‘“(𝑧𝑀) for every π‘“βˆˆπΈπ‘§π‘€. From this by induction we have equality (2.14) (we omit details).
Then we have ξ€·π‘Šπ‘˜π‘§π‘€π‘“ξ€Έ(𝑧𝑀)=(𝑧𝑀)π‘˜[]π‘˜!2πœ•βŠ›π‘“(𝑧𝑀)=2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜[](π‘˜!)2=1𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒(π‘˜!)2ξ€œπ‘§0ξ€œπ‘€0π‘˜2[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜βˆ’1𝑓=π‘˜(𝑒𝑣)𝑑𝑣𝑑𝑒2π‘˜2[](π‘˜βˆ’1)!2ξ€œπ‘§0ξ€œπ‘€0[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜βˆ’1=ξ€œπ‘“(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧0ξ€œπ‘€0[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜βˆ’1[](π‘˜βˆ’1)!2𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒,(2.15) which proves (2.13).
Now by combining (2.9) and (2.13) we have 𝑇(𝑧𝑀)π‘˜=π‘˜ξ‘π‘ =1𝑠2ξ€œπ‘§0ξ€œπ‘€0[](π‘§βˆ’π‘’)(π‘€βˆ’π‘£)π‘˜βˆ’1[](π‘˜βˆ’1)!2𝑇1𝑑𝑣𝑑𝑒,(2.16) for all π‘˜β‰₯0, which means that 𝑇(𝑧𝑀)π‘˜=(𝑧𝑀)π‘˜βŠ›π‘‡1(π‘˜β‰₯0),(2.17) and hence 𝑇𝑝(𝑧𝑀)=𝑝(𝑧𝑀)βŠ›π‘‡1,(2.18) for all polynomials 𝑝. Thus, by Lemma 2.1 and Weierstrass approximation theorem, we deduce that πœ•(𝑇𝑓)(𝑧𝑀)=𝑇1βŠ›π‘“(𝑧𝑀)=2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0=πœ•π‘“((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑇1)(𝑒𝑣)𝑑𝑣𝑑𝑒2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0=ξ€œ(𝑇1)((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧0ξ€œπ‘€0πœ•2=ξ€œπœ•π‘§πœ•π‘€(𝑇1)((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧0ξ€œπ‘€0ξ€Ί(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)(𝑇1)𝑧𝑀((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(𝑇1)𝑀=ξ€œ((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒+(𝑇1)(0)𝑓(𝑧𝑀)𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒+πœ‘0𝑓(𝑧𝑀),(2.19) where πœ‘βˆΆ=𝑇1βˆˆπΈπ‘§π‘€.(2.20) Thus, (𝑇𝑓)(𝑧𝑀)=πœ‘(𝑧𝑀)βŠ›π‘“(𝑧𝑀)=π’Ÿπœ‘π‘“(𝑧𝑀)(2.21) for all π‘“βˆˆπΈπ‘§π‘€ and some πœ‘βˆˆπΈπ‘§π‘€. Conversely, if πœ‘βˆˆπΈπ‘§π‘€, then π’Ÿπœ‘ commutes with π‘Šπ‘§π‘€. Since by Lemma 2.1𝐸𝑧𝑀 is a Banach algebra with respect to the Duhamel product βŠ›, π’Ÿπœ‘ is bounded operator on 𝐸𝑧𝑀. The theorem is proved.

Corollary 2.3. For a function πœ‘ in 𝐸𝑧𝑀 there exists a unique commutant 𝑇 of the operator π‘Šπ‘§π‘€ such that 𝑇1=πœ‘(𝑧𝑀).

Corollary 2.4. One has {π‘Šπ‘§π‘€}ξ…žξ…ž={π‘Šπ‘§π‘€}ξ…ž, where {π‘Šπ‘§π‘€}ξ…žξ…ž stands for the bicommutant of the operator π‘Šπ‘§π‘€.

Proof. It suffices to prove that 𝑇1𝑇2=𝑇2𝑇1 for every 𝑇1, 𝑇2 in {π‘Šπ‘§π‘€}ξ…ž. Indeed, by Theorem 2.2, there exist πœ‘,πœ“βˆˆπΈπ‘§π‘€ such that 𝑇1π‘“ξ€Έξ€œ(𝑧𝑀)=πœ‘(0)𝑓(𝑧𝑀)+𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€+((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»π‘“=ξ€·((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑒𝑣)π‘‘π‘£π‘‘π‘’πœ‘(0)𝐼+𝒦Φ𝑇𝑓(𝑧𝑀),(2.22)2π‘“ξ€Έξ€œ(𝑧𝑀)=πœ“(0)𝑓(𝑧𝑀)+𝑧0ξ€œπ‘€0ξ€Ίπœ“π‘€+((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ“π‘§π‘€ξ€»π‘“=ξ€·((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑒𝑣)π‘‘π‘£π‘‘π‘’πœ“(0)𝐼+𝒦Ψ𝑓(𝑧𝑀)(2.23) for all π‘“βˆˆπΈπ‘§π‘€, where πœ•Ξ¦(𝑧𝑀)∢=2=πœ•πœ•π‘§πœ•π‘€πœ‘(𝑧𝑀),Ξ¨(𝑧𝑀)∢2π’¦πœ•π‘§πœ•π‘€πœ“(𝑧𝑀),Ξ¦ξ€œπ‘“(𝑧𝑀)=(Ξ¦βˆ—π‘“)(𝑧𝑀)∢=𝑧0ξ€œπ‘€0πœ•2=ξ€œπœ•π‘§πœ•π‘€πœ‘((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»π’¦((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒,Ξ¨π‘“ξ€œ(𝑧𝑀)=(Ξ¨βˆ—π‘“)(𝑧𝑀)∢=𝑧0ξ€œπ‘€0πœ•2πœ“=ξ€œπœ•π‘§πœ•π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧0ξ€œπ‘€0ξ€Ίπœ“π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ“π‘§π‘€ξ€»π‘“((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑒𝑣)𝑑𝑣𝑑𝑒.(2.24) Since the usual convolution operators 𝒦Φ and 𝒦Ψ are commuting operators, we have 𝑇1𝑇2=ξ€·πœ‘(0)𝐼+π’¦Ξ¦ξ€Έξ€·πœ“(0)𝐼+𝒦Ψ=ξ€·πœ“(0)𝐼+π’¦Ξ¨ξ€Έξ€·πœ‘(0)𝐼+𝒦Φ=𝑇2𝑇1,(2.25) which proves the corollary.

Theorem 2.5. An operator π‘‡βˆˆβ„’(𝐸𝑧𝑀) will be an isomorphism of the space 𝐸𝑧𝑀 into itself and commutes with π‘Šπ‘§π‘€ if and only if it can be written in the form ξ€œ(𝑇𝑓)(𝑧𝑀)=πœ‘(0)𝑓(𝑧𝑀)+𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€+((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»π‘“((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑒𝑣)𝑑𝑣𝑑𝑒,(2.26) andπœ‘(0)=(𝑇1)βˆ£π‘§π‘€=0β‰ 0.

Proof. If π‘‡βˆˆβ„’(𝐸𝑧𝑀) is an isomorphism of the space 𝐸𝑧𝑀 into itself and commutes with π‘Šπ‘§π‘€, then by Theorem 2.2 we have for 𝑇 representation (2.26) with π‘‡π‘Šπ‘§π‘€=π‘Šπ‘§π‘€π‘‡. Clearly, it follows from this equality and (2.26) that πœ‘(0)=(𝑇1)βˆ£π‘§π‘€=0β‰ 0.
Conversely, suppose that 𝑇 has the form (2.26) with πœ‘(0)=(𝑇1)βˆ£π‘§π‘€=0β‰ 0, and prove then that π‘‡βˆˆ{π‘Šπ‘§π‘€}ξ…ž and 𝑇 is an isomorphism on 𝐸𝑧𝑀. Indeed, the inclusion π‘‡βˆˆ{π‘Šπ‘§π‘€}ξ…ž follows directly from Theorem 2.2. On the other hand, it is easy to see from the representation (2.26) that 𝑇=π’Ÿπœ‘=πœ‘(0)𝐼+𝒦Φ,(2.27) where 𝒦Φ, π’¦Ξ¦ξ€œπ‘“=𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒,(2.28) is the usual convolution operator on 𝐸𝑧𝑀. It is not difficult to see that 𝒦Φ is a compact operator on 𝐸𝑧𝑀.
Let us show that kerπ’Ÿπœ‘={0}. Indeed, let π’Ÿπœ‘π‘“=0, where π‘“βˆˆπΈπ‘§π‘€. Then, (πœ‘(0)𝐼+𝒦Φ)𝑓=0, that is, πœ•2ξ€œπœ•π‘§πœ•π‘€π‘§0ξ€œπ‘€0πœ‘((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒=0.(2.29) By standard calculation, we obtain from (2.29) that ξ€œπ‘§0ξ€œπ‘€0πœ‘((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒=𝑐1𝑧+𝑐2,(2.30) where 𝑐1, 𝑐2 are constants. Since ξ‚΅ξ€œ0=𝑧0ξ€œπ‘€0πœ‘ξ‚Ά||||((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑧=0=𝑐1𝑧+𝑐2ξ€Έ||𝑧=0=𝑐2,(2.31) we have that 𝑐2=0. On the other hand, since ξ‚΅ξ€œ0=𝑧0ξ€œπ‘€0πœ‘ξ‚Ά||||((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒𝑀=0=𝑐1𝑧,(2.32) for all π‘§βˆˆπ”», we obtain that 𝑐1=0. Thus, ξ€œπ‘§0ξ€œπ‘€0πœ‘((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))𝑓(𝑒𝑣)𝑑𝑣𝑑𝑒=0,(2.33) for all π‘§βˆˆπ”» and π‘€βˆˆπ”». Now, by considering that πœ‘(0)β‰ 0 and πœ‘ is a continuous function on 𝔻×𝔻, by the Titchmarsh Convolution Theorem [17] for functions of several variables we deduce from (2.33) that 𝑓(𝑧𝑀)=0 for all 𝑧,π‘€βˆˆπ”», that is, ker𝑇={0}. Since 𝒦Φ is compact, it follows from Fredholm alternative that 𝑇 is invertible in 𝐸𝑧𝑀, that is, 𝑇 is an isomorphism. The theorem is proved.

From Theorem 2.5 and Corollary 2.3 we obtain the following.

Corollary 2.6. For any function πœ‘ belonging to 𝐸𝑧𝑀 and satisfying πœ‘(0)β‰ 0, there exists a unique isomorphism 𝑇 of the space 𝐸𝑧𝑀 such that 𝑇 commutes with π‘Šπ‘§π‘€ and 𝑇1=πœ‘(𝑧𝑀).

Corollary 2.7. If πœ‘βˆˆπΈπ‘§π‘€ and πœ‘(0)β‰ 0, then the integrodifferential equation ξ€œπœ‘(0)π‘₯(𝑧𝑀)+𝑧0ξ€œπ‘€0ξ€Ίπœ‘π‘€((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))+(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)πœ‘π‘§π‘€ξ€»((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))π‘₯(𝑒𝑣)𝑑𝑣𝑑𝑒=𝑦(𝑧𝑀)(2.34) has a unique solution for any right-hand side π‘¦βˆˆπΈπ‘§π‘€.

Corollary 2.8. The unique maximal ideal of the Banach algebra (𝐸𝑧𝑀,βŠ›) is {π‘“βˆˆπΈπ‘§π‘€βˆΆπ‘“(0)=0}; that is, the maximal ideal space of (𝐸𝑧𝑀,βŠ›) consists of one homomorphizm, namely evaluation at the origin β„Ž(𝑓)=𝑓(0).

3. Cyclic Vectors of π‘Šπ‘§π‘€

Let us consider the restricted operator π‘Šπ‘§π‘€=π‘Šβˆ£π»π‘π‘§π‘€. In this section we will describe the set of all cyclic vectors of this operator. The main result of this section is the following.

Theorem 3.1. Let π‘“βˆˆπ»π‘π‘§π‘€. Then ξ€½π‘Šspanπ‘›π‘§π‘€ξ€Ύπ‘“βˆΆπ‘›=0,1,2,…=𝐻𝑝𝑧𝑀,(3.1) if and only if π‘“βˆ£π‘§π‘€=0β‰ 0.

Proof. It is easy to verify that π‘Šπ‘˜π‘§π‘€π‘”(𝑧𝑀)=(𝑧𝑀)π‘˜(π‘˜!)2βŠ›π‘”(𝑧𝑀)(π‘˜β‰₯0),(3.2) for all π‘”βˆˆπ»π‘π‘§π‘€. Let us define the integrodifferential operator (or, briefly, the Duhamel operator) π’Ÿπ‘“ defined by (π’Ÿπ‘“π‘”)(𝑧𝑀)=(π‘“βŠ›π‘”)(𝑧𝑀),π‘”βˆˆπ»π‘π‘§π‘€. By the known result of Merryfield and Watson (see [3, Corollary 2.6]), 𝐻𝑝(𝔻2),𝑝β‰₯1, is the Banach algebra with respect to the Duhamel product βŠ› defined by (1.1). Therefore, it is easy to see that (𝐻𝑝𝑧𝑀,βŠ›) is also Banach algebra, and hence, π’Ÿπ‘“ is a bounded operator on 𝐻𝑝𝑧𝑀. Then it follows from (3.2) that ξ€½π‘Šspanπ‘˜π‘§π‘€ξ€Ύξ‚»π‘“βˆΆπ‘˜β‰₯0=span(𝑧𝑀)π‘˜(π‘˜!)2ξ‚Όξ‚»π’ŸβŠ›π‘“(𝑧𝑀)βˆΆπ‘˜β‰₯0=span𝑓(𝑧𝑀)π‘˜(π‘˜!)2ξ‚Άξ‚ΌβˆΆπ‘˜β‰₯0=closπ’Ÿπ‘“ξ€½span(𝑧𝑀)π‘˜ξ€ΎβˆΆπ‘˜β‰₯0=closπ’Ÿπ‘“π»π‘π‘§π‘€.(3.3) Thus, 𝑓 is a cyclic vector for π‘Šπ‘§π‘€ if and only if π’Ÿπ‘“ has a dense range. Let us show that the latter is equivalent to the condition 𝑓(0)β‰ 0. Clearly, if π’Ÿπ‘“ has a dense range then 𝑓(0)β‰ 0. Conversely, let 𝑓(0)β‰ 0. We will prove actually more strong result that π’Ÿπ‘“ is invertible in 𝐻𝑝𝑧𝑀. Really, let us rewrite the operator π’Ÿπ‘“ in the form π’Ÿπ‘“=𝑓(0)𝐼+π’¦πœ•2𝑓/πœ•π‘§πœ•π‘€, where 𝐼 is the identity operator in 𝐻𝑝𝑧𝑀 and π’¦πœ•2𝑓/πœ•π‘§πœ•π‘€ξ€œπ‘”(𝑧𝑀)=𝑧0ξ€œπ‘€0𝑓𝑧+((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(π‘§βˆ’π‘’)(π‘€βˆ’π‘£)𝑓𝑧𝑀𝑔((π‘§βˆ’π‘’)(π‘€βˆ’π‘£))(𝑒𝑣)𝑑𝑣𝑑𝑒.(3.4) Since πœ•2𝑓/πœ•π‘§πœ•π‘€ is a continuous function, it is easy to see that π’¦πœ•2𝑓/πœ•π‘§πœ•π‘€ is a compact operator (even Volterra operator) on 𝐻𝑝𝑧𝑀. Now, as in the proof of Theorem 2.5, it follows from Titchmarsh Convolution Theorem that kerπ’Ÿπ‘“={0}. Then, again by the Fredholm theorem we assert that π’Ÿπ‘“ is invertible, which completes the proof.

In conclusion, note that the study of the double integration operator π‘Š in the Lebesgue space 𝐿2([0,1]Γ—[0,1]) was originated by Donoghue, Jr., in [18]. He showed that the operator π‘Š is not unicellular. Atzmon and Manos [19] proved that the multiplicity of spectrum πœ‡(π‘Š) of the operator π‘Š is equal to +∞ (we recall that the multiplicity of spectrum of the Banach space operator π΄βˆˆβ„’(𝑋) is defined by πœ‡(𝐴)∢=min{card𝐸∢span{π΄π‘›πΈβˆΆπ‘›β‰₯0}=𝑋}). Some related results for π‘Š are also contained in the paper [15] by Karaev.

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Copyright © 2012 Suna Saltan and Yasemin Özel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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