Gen. Math. Notes, Vol. 17, No. 1, July 2013, pp. 1-7 ISSN 2219-7184; Copyright c ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in
Topological Results About the Set of Generalized Rhaly Matrices
Nuh Durna1 and Mustafa Yildirim2
1,2 Cumhuriyet University, Faculty of Sciences Department of Mathematics, 58140 Sivas, Turkey 1 E-mail: [email protected] 2 E-mail: [email protected]; [email protected]
(Received: 1-2-12 / Accepted: 15-5-13)
Abstract In this paper, we will investigate some topological properties of Rb set of all bounded generalized Rhally matrices given by b sequences in H2. We will show 2 that Rb is closed subspace of B (H ) and we will define an operator from T to Rb and show that this operator is injective and continuous in strong-operator topology. Keywords: Generalized Rhaly operators, weak-operator topology, strong- operator topology.
1 Introduction
A Hilbert space has two useful topologies (weak and strong); the space of operators on a Hilbert space has several. The metric topology induced by the norm is one of them; to distinguish it from the others, it is usually called the norm topology or the uniform topology. The next two are natural outgrowths for operators of the strong and weak topologies for vectors. A subbase for the strong operator topology is the collection of all sets of the form
{A : k(A − A0) fk < } ; correspondingly a base is the collection of all sets of the form
{A : k(A − A0) fik < , i = 1, 2, . . . , k} 2 Nuh Durna et al.
Here k is a positive integer, f1, f2, . . . , fk are vectors, and is a positive number. A subbase for the weak operator topology is the collection of all sets of the form
{A : |h(A − A0) f, gi| < } , where f and g are vectors and > 0; as above (as always) a base is the collection of all finite intersections of such sets. The corresponding concepts of convergence (for sequences and nets) are easy to describe: An → A strongly if and only if Anf → Af strongly for each f (i.e., k(An − A) fk → 0 for each f), and An → A weakly if and only if Anf → Af weakly for each f (i.e., hAnf, gi → hAf, gi for each f and g). For a slightly different and often very efficient definition of the strong and weak operator topologies (see [3, Problems 224 and 225]). The easiest questions to settle are the ones about comparison. The weak topology is smaller (weaker) than the strong topology, and the strong topology is smaller than the norm topology. In other words, every weak open set is a strong open set, and every strong open set is norm open. In still other words: every weak neighborhood of each operator includes a strong neighborhood of that operator, and every strong neighborhood includes a metric neighborhood. Again: norm convergence implies strong convergence, and strong convergence implies weak convergence. These facts are immediate from the definitions. In the presence of uniformity on the unit sphere, the implications are reversible (see [3]). Let H (D) denotes the space of complex valued analytic function on the unit disk D, for 1 ≤ p < ∞, Hp denotes the standard Hardy space on D. Integral operators have been studied of the greater part of the last century. In [5], Scott W. Young gave definition of generalized Ces`aromatrix as follows:
Definition 1.1 Let g be analytic on the unit disk. The generalized Ces`aro operator on H2 with symbol g
z 1 Z C (f)(z) = f (t) g (t) dt (1) g z 0
In [4], Pommerenke defined the following operator on H2 :
z Z 0 IG (f) = f (t) G (t) dt. 0
He showed the following lemma: Topological Results About the Set... 3
2 Lemma 1.2 ([4, Lemma 1]) IG is bounded on H iff G ∈ BMOA. Mor- ever, there is a constant K1 such that kIGkop ≤ K1 kGk∗.
z R If G (z) = g (w) dw, then Pommerenke [4] showed that Cg is bounded 0 on the Hilbert space H2 if and only if G ∈ BMOA. Aleman and Siskakis [1] extended Pommerenke’s result to the Hardy spaces Hp for all p, 1 ≤ p < ∞, p and show that Cg is compact on H if and only if G ∈ VMOA. In [2], the authors defined generalized Rhaly matrix by using the general- ized Ces`aromatrix as following:
∞ P k Definition 1.3 Let {bn} be a scalar sequence and g (z) = akz ∈ H (D). k=0 The matrix a0b0 a1b1 a0b1 g R = a2b2 a1b2 a0b2 (2) b a b a b a b a b 3 3 2 3 1 3 0 3 . . . . . ...... is called generalized Rhaly matrix with symbol g on H2.
From (2), we can write an−jbn , n ≥ j g (Rb )nj = . (3) 0 , n < j
g The relation Rb = DbCg is valid similar to on Rhaly matrix, where Db = ∞ 1 diag {(n + 1) bn}n=0. We recall that Cg = C1 for g (z) = 1−z . Since g (z) = ∞ P k z , which fixes then an = 1 for all n ∈ N. Thus, from (2) we get k=0 b0 b1 b1 g R = b2 b2 b2 = R . b b b3 b3 b3 b3 . . . . . ...... 4 Nuh Durna et al.
g On the other hand R1/(n+1) = Cg. Therefore this definition could be regarded as a two-way generalizing both for Rhaly and Ces`arooperator. We define the following set B: Z z B := g : D → C : g (t) dt ∈ BMOA . 0 The authors gave for bounded of generalized Rhally matrices as follows:
Lemma 1.4 ([2, Lemma 3.1]) If g ∈ B and {(n + 1) bn} is bounded se- g quence, then Rb is bounded.
For this paper, we make the convention that whenever β or βj is notated, β 1/(1−βz) it is assumed that |β| = |βj| = 1. Furthermore, Rb = Rb . The authors gave for unitarily equivalent of generalized Rhally matrices as follows:
gβ Lemma 1.5 ([2, Theorem 5]) If gβ (z) = g (βz) with |β| = 1, then Rb g gβ ∼ g is unitarily equivalent to Rb . We denote this by Rb = Rb .
2 Topological Results of Generalized Rhaly Ma- trices
In this part, we will investigate some topological properties of Rb set of all bounded generalized Rhally matrices given by b sequences in H2. From Lemma 1.4, Rb is g Rb = {Rb : g ∈ B} where {(n + 1) bn} is bounded sequence. We will show that Rb is closed sub- space of B (H2) and we will define an operator from T := {z : kzk = 1} to Rb and show that this operator is injective and continuous in strong-operator topology.
Definition 2.1 The weak-operator topology, or WOT, on B (H) is the lo- cally convex topology generated by the seminorms, Tx,y = |hT x, yi| for x, y ∈ H and T ∈ B (H).
WOT Equivalently, if {Tα} is a net of operators in B (H); then, Tα → T iff hTαx, yi → hT x, yi for each x, y ∈ H.
Definition 2.2 The strong-operator topology, or SOT, on B (H) is the lo- cally convex topology generated by the seminorms, Tx = kT xk for x ∈ H and T ∈ B (H). Topological Results About the Set... 5
As with WOT-convergence, SOT-convergence can be formulated as follows: SOT if {Tα} is a net of operators in B (H); then, Tα → T iff k(Tα − T ) xk → 0 for each x ∈ H.
2 Theorem 2.3 Rb is a Weak-Operator closed subspace of B (H ).
g1 g2 g1 g2 g1+g2 Proof Let Rb ,Rb ∈ Rb. Then Rb + Rb = Rb ∈ Rb since g1 + g2 ∈ B for all g1, g2 ∈ B. We prove that Rb is closed in the weak operator topology. g g WOT Let {Rb }α be a net of operators in Rb such that {Rb }α → S. We must h P∞ k show that S = Rb for some h ∈ B. Let k=0 (ak)α z be the Taylor series for 2 2 g gα. S ∈ B (H ) since B (H ) is WOT-closed. Let {Rb }α nj be the matrix for g 2 {Rb }α in the standart basis for H . g WOT {Rb }α → S. Thus, we get g {Rb }α nj → Snj for each n, j. We know that for any bounded operator on a Hilbert space T , the nj−th ∞ entry in a matrix for T in an orthonormal basis {en}n=1 is Tnj = hT en, eji. g {Rb }α nj = (an−j)α bn if n ≥ j. This implies that Snj = limα (an−j)α bn for g n ≥ j. For n < j, we know that {Rb }α nj = 0. Therefore, Snj = 0 for n < j. So S has the form of (3). Define ∞ X g n h (z) := c + {Rb }α n1 z , n=1 h where c ∈ C. Then S = Rb . By Lemma 1.2, h ∈ B . Thus S ∈ Rb. The WOT is weaker than the norm topology, thereby giving us that Rb is 2 also norm closed. Therefore Rb is closed subspace of B (H ). Now, we want to define a map from T to Rb. If we let β ∈ T, then define
1 1−βz φ : T → Rb , β → Rb .
Theorem 2.4 φ : T → Rb is a strong-operator continuous bijective map.
Proof Let βk → β. We must show that
1 1 1−β z 1−βz R k − R (f) → 0 b b
∞ 2 P n for each f ∈ H . Let f (z) = anz . From Lemma 1.5, we have n=0
1 1 1 1−βkz 1−βz ∼ 1−ββkz Rb − Rb = Rb − Rb. 6 Nuh Durna et al.
If f ∈ H2, then
1 1 1 1−ββkz 1−βkz 1−βz R − Rb (f) = Uβ R − R U (f) b b b B 1 1 1−β z 1−βz = R k − R U (f) b b B 1 1 1−β z 1−βz = R k − R (g) b b 2 where g = UB (f) ∈ H . Thus,
1 1 1 1−βkz SOT 1−βz 1−ββkz SOT Rb → Rb ⇔ Rb → Rb. We may assume β = 1.
1 ∞ n ! 1−βkz X X n Rb (f (z)) = bn aj (βkz) n=0 j=0 ∞ n ! X X n Rb (f (z)) = bn aj z . n=0 j=0 Comparing these last equations, we get
1 ∞ n ! 1−βkz X X j Rb − Rb (f) = bn aj βk − 1 . n=0 j=0
j−1 j 2 j−1 P r Since βk −1 = (βk − 1) 1 + βk + βk + ··· + βk = (βk − 1) βk, we obtain r=0 that 2 1 2 ∞ n 1−βkz X 2 X j R − Rb (f) = b aj β − 1 b n k n=0 j=0 j−1 2 ∞ n X 2 2 X X r = bn |βk − 1| aj βk n=0 j=0 r=0 j−1 2 ∞ n 2 X X X r = |βk − 1| ajbn βk . n=0 j=0 r=0
1 1−βkz 2 Rb − Rb (f) ∈ H implies that for every > 0, there is an A such that for every n ≥ A, we have
∞ n j−1 2 2 X X X |β − 1| a b βr < . (4) k j n k 2 n=A j=0 r=0 Topological Results About the Set... 7
Let j−1 2 A−1 n X X X r B = ajbn βk . n=0 j=0 r=0 Since sum is finite, B < ∞. β → 1 means that for every > 0, there is a K k such that if k ≥ K, then |β − 1| < . Fix > 0. If we take n ≥ A and k 2B k ≥ K, then we obtain that
j−1 2 j−1 2 ∞ n A−1 n 2 X X X r 2 X X X r |βk − 1| ajbn βk = |βk − 1| ajbn βk n=0 j=0 r=0 n=0 j=0 r=0 j−1 2 ∞ n 2 X X X r + |βk − 1| ajbn βk n=A j=0 r=0 < .B + = . 2B 2
1 1−ββkz SOT So, Rb → Rb. Therefore, φ, SOT-continuous.
References
[1] A. Aleman and A.G. Siskakis, An integral operator on Hp, Complex Vari- ables, 28(1995), 149-158.
[2] N. Durna and M. Yildirim, Generalized Rhaly matrices, (In Review).
[3] P.L. Halmos, A Hilbert Space Problem Book (Second Edition), Springer- Verlag, New York, Heidelberg, Berlin, (1982).
[4] Ch. Pommerenke, Schlichte funktionen und analytische funktionen von beschr¨anktermittlerer oszillation, Comment. Math. Helv., 52(1977), 591- 602.
[5] S.W. Young, Spectral properties of generalized Ces`arooperators, Integr. Equ. Oper. Theory, 50(2004), 129-146.