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On the spectrum and Hilbert Schimidt properties of generalized Rhaly matrices Article  in  Communications · April 2018 DOI: 10.31801/cfsuasmas.464177

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Com mun. Fac. Sci. Univ. Ank. Ser. A1 M ath. Stat. Volum e 68, Numb er 1, Pages 712–723 (2019) DOI: 10.31801/cfsuasm as.464177 ISSN 1303–5991 E-ISSN 2618-6470 http://com munications.science.ankara.edu.tr/index.php?series=A1

ON THE SPECTRUM AND HILBERT SCHIMIDT PROPERTIES OF GENERALIZED RHALY MATRICES MOHAMMAD MURSALEEN, MUSTAFA YILDIRIM, AND NUH DURNA Abstract. In this paper, we show that the multiplications of some generalized Rhaly operators are Hilbert Schimidt operators.We also calculate the spectrum and essential spectrum of a special generalized Rhaly matrices on the Hardy space H 2 .

1. Introduction and preliminaries Let H (D) denotes the space of complex-valued analytic functions on the unit disk D. Let H p (1 p < 1) denote the standard Hardy space on D, and `p denote the standard space of p-summable complex-valued sequences on the set of non-negative integers. In [15], A.G. Siskakis gave the spectrum of Cesàro matrix on H p by using the integral representation of Cesàro operator. The Cesàro operator C is a paradigm of a noncompact P1 operator in this class. Recall that C has the following form on H 2 : If f (z) = n=0 an z n 2 H 2 , then ! n 1 X X 1 C (f ) (z) = ai z n : (1) n + 1 n=0 i=0 We get the following integral representation of (1) by calculating the Taylor series Z 1 z f (t) C (f ) (z) = dt: (2) z 0 1 t In [19], S.W. Young generalized the Cesàro operator, by taking more general analytic function instead of the function 1= (1 t) in equality (2) as follows:

Received by the editors: February 09, 2018; Accepted: April 11, 2018. 2010 Mathematics Subject Classi…cation. Primary 47A10; Secondary 47G10, 32A35, 47B99. Key words and phrases. Cesàro operators, terraced operators, Rhaly operators, generalized Cesàro operators, generalized terraced operators, spectrum, essential spectrum, Hilbert Schimidt. c 2 0 1 8 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rsity o f A n ka ra -S e rie s A 1 M a th e m a tic s a n d S ta tistic s

712

GENERALIZED RHALY M ATRICES

713

De…nition 1. Let g be an analytic function on the unit disk. The generalized Cesàro operator on H 2 with symbol g is de…ned by Z 1 z Cg (f ) (z) = f (t) g (t) dt: (3) z 0

De…nition Z 2. Let I be an arc of the unit circle T, and let ' : T ! C. Then, let 1 'I = j'j ; where jIj denotes the arclength of I. ' is said to be of bounded jIj I mean oscillation if Z 1 k'k = sup j' 'I j < 1: I T jIj I We denote the set of all functions of bounded mean oscillation by BM O. If we endow BM O with the norm k'kBM O = k'k + j' (0)j, then BM O is a Banach space (see [8]). We say that g 2 BM OA if g 2 H 2 and g ei 2 BM O.

De…nition 3. Let I be an arc of T. We say that a function ' : T ! C is of vanishing mean oscillation if Z 1 j' 'I j = 0: lim sup !0 I jIj I

We denote the set of all functions of vanishing mean oscillation by V M O. V M O is a closed subspace of BM O: As with BM OA, we de…ne V M OA as the set of g 2 H 2 such that g ei 2 V M O. V M OA is a closed subspace of BM OA (see [8]). We denote the spectrum of the linear operator T by Rz

(T ) = f 2 C : T

(T ). That is,

not invertibleg :

If G (z) = 0 g (w) dw; then Pommerenke [12] showed that Cg is bounded on the Hilbert space H 2 if and only if G 2 BM OA: Aleman and Siskakis [2] extended Pommerenke’s result to the Hardy spaces H p for 1 p < 1; and show that Cg is compact on H p if and only if G 2 V M OA. In [10], Hardy, Littlewood, and Polya showed that C is bounded on `p . In [3], Brown, Halmos and Shields obtained that the spectrum for C on `p , is in the form (C) = fz : jz 1j 1g : Gonzalez [9] proved that e (C; `p ) = fz : jz 1j = 1g. Spectrums of C Cesàro matrix on some other sequence spaces are taken consideration in some studies like [11] and [1]. Given a sequence a = (an ) of scalers, a terraced matrix Ra = (ank ) is the lower triangular matrix de…ned by ank = an , k n and ank = 0, otherwise. In [17], Yildirim calculated the spectrum of terraced matrices on `p , such that if 0 6= L < 1 then (Ra ) = fz : jz qL=2j qL=2g for p > 1 and p 1 + q 1 = 1, where L = limn (n + 1) an and S = fan : n = 0; 1; 2; : : :g. Later Valeryevna [16] showed that essentially spectrum of terraced matrices on `p is e (Ra ) = f : j qL=2j = qL=2g, where 1 < p < 1; 0 6= L = limn (n + 1) an and q = p= (p 1). If entries of Ra terraced matrix are taken as an = 1= (n + 1),

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M OHAM M AD M URSALEEN, M USTAFA YILDIRIM , AND NUH DURNA

Cesàro matrix will be obtained and the spectrum results for this special terraced matrix will con‡ict with the previous results. A lower triangular matrix A is said to be factorable if ank = an bk for all 0 k n. Spectrum of the factorable matrix on `p is studied in [14]. The choices an = 1=(n + 1) and each bk = 1, an = an and each bk = 1 generate C (the Cesàro matrix of order one), terraced matrices de…ned 2 by Rhaly [13], respectively. In [18], the …ne spectrum Pn for aCi g on H was computed when g is a rational symbol such that, if g (z) = i=1 1 z , where j i j = 1, i ’s i are distinct, and ai 6= 0 for each i; then for Cg ; Sn Sn 1 (a) (Cg ) = Si=1 D (ai ) [ F where F = fg (0) =kgk=1 n i=1 D (ai ), n (b) e (Cg ) = i=1 @D (ai ), Pn (c) For 2 = e (Cg ), ind (Cg ) = G ( ) where G = i=1 D(ai ) and D (a) = fz : jz aj < jajg for any a 2 C. Furthermore, certain products of generalized Cesàro operators are shown to be a Hilbert-Schimidt operator. In the next de…nition we will give generalized terraced matrix by using the generalized Cesàro matrix given in [7]. P1 De…nition 4. Let fbn g be a scalar sequence and g (z) = k=0 ak z k 2 H (D). The matrix 0 1 a0 b0 B C B C B a1 b1 C a b 0 1 B C B C B C B a2 b2 C g a b a b 1 2 0 2 (4) Rb = B C B C B C B a3 b3 C a 2 b3 a1 b3 a0 b3 B C B C @ A .. .. .. .. .. . . . . . is called generalized terraced matrix with symbol g on H 2 . The relation Rbg = Db Cg is valid similar to on Rhaly matrix, where Db = 1 1 diag f(n + 1) bn gn=0 . We recall that Cg = C1 for g (z) = : Since g (z) = 1 z P1 k k=0 z , which …xes then an = 1 for all n 2 N. Thus, from (4), we get 0 1 b0 B b1 C b1 B C B b2 C g b b 2 2 Rb = B C = Rb : B b3 C b b b 3 3 3 @ A .. .. .. .. .. . . . . .

g On the other hand R1=(n+1) = Cg . Therefore this de…nition could be regarded as a two-way generalizing both for Rhaly and Cesàro operator. In [6], Durna and Yildirim have investigated some topological results about the set of generalized Rhally matrices.

GENERALIZED RHALY M ATRICES

715

For this paper, we make the convention that whenever assumed that j j =

j

= 1. Furthermore, Rb =

1=(1 Rb

or

z)

j

is notated, it is

.

2. Hilbert-Schimidt We used the lemmas in the Young’s paper [20] to provide the spectrum of generalized terraced matrices. We have showed that Rb 1 Rb 2 , Rb 1 Rb 2 and Rb 1 Rb 2 are Hilbert-Schmidt operator as 1 6= 2 , to enable the hypothesis of Young’s lemmas in this section. Pn Lemma 5. If 2 Z, Bj 1 = 0 , Bk = k=j k for k = j; : : : ; n and (ak ) is a positive decreasing sequence, then ! n k X 2 1 1 1 : + ak j1 j an aj an k=j

Proof. For each k, jBk j =

n X

k

j

=

n j+1

1

j

1

k=j

1+ j1

n j+1

j

=

Then, we have n X

k

ak

=

k=j

=

n X

1 ak

(Bk

k=j n X

Bk ak

Bn an

+

n X1

Bk ak+1

Bk

1 ak

1)

k=j

k=j

=

Bk

n X1

1 ak+1

:

k=j

If we take absolute values in the last equations, we have n X k=j

n 1

k

ak

=

Bn X + Bk an k=j

n 1

1 ak

1 ak+1

X Bn 1 1 + Bk an ak ak+1 k=j 0 1 n X1 1 2 @ 1 1 A + j1 j an ak ak+1 k=j ! 1 1 2 1 + : j1 j an aj an

2 j1

j

.

716

M OHAM M AD M URSALEEN, M USTAFA YILDIRIM , AND NUH DURNA

Theorem 6. Let (bn ) 2 `2 is a positive decreasing sequence. Then Rb 1 Rb 2 is a Hilbert-Schmidt operator if 1 6= 2 . Proof. Rb 1 Rb 2 2 B2 H 2 , U 2 Rb 1 U 2 U 2 Rb 2 U 2 2 B2 H 2 . Since, U 2 Rb 2 U 2 = Rb by [7, Theorem 5] then, we can assume 2 = 1. Rewrite 1 as . Since 1=(1 z) Rb = Rb ; n j

Rb bn 0

Since (Rb )nk =

Rb Rb

nj

; ; =

nj

bn 0

=

; ;

1

n j n : 0 8 n X > n < bn 1 = k=j > : 0

k

k

bk

bk

1

1

;

n

j

;

n 0 such that j(n + 1) bn j for each n 2 N. Hence, we get 1 1 X X 2 2 2 2 j(n + 1) bn j jxn j M2 jxn j = M 2 kxk , n=0

M

n=0

i.e.,

kDb xn k M kxk . Thus Db is bounded. Furthermore, since Cg is bounded for g 2 B, Rbg = Db Cg is bounded. Remark 10. If we take bn = 1= (n + 1) in Lemma 9, we obtain [19, expression in page 130]. Lemma 11 ([20, Lemma 3.2]). If a1 ; a2 ; : : : ; an are elements of a C -algebra A such that ai aj = aj ai = 0 and ai aj = aj ai = 0 for i 6= j and i 2 C, then ! n n X [ f0g [ = ( i ai ) : (10) i ai i=1

i=1

We note that, the f0g in (10) is necessary because the sum of non-invertible elements can be invertible. For example, let P be a projection; P P ? = P ? P = 0, but P + P ? = I. Lemma 12 ([20, Lemma 3.3]). IfST1 ; : : : ; Tn 2 B (H) such that Ti Tj is compact n for i 6= j and i 2 C, then for 2 = i=1 e (Ti ), we have that ! n n X X ind T = ind ( i Ti ): (11) i i i=1

i=1

Lemma 13 ([16, Theorem 2.2]). Let 1 < p < 1; 0 6= L = limn (n + 1) an and q = p= (p 1) ; then qL qL = : (12) : e (Ra ) = 2 2 The …nal lemma is well-known and can easily be derived. Lemma 14. If T is a bounded linear operator on a Hilbert space such that it has a lower triangular matrix representation in an orthonormal basis B with diagonal 1 1 entries f n gn=0 , then p (T ) f n gn=0 . Theorem 15. Let (bk ) 2 `2 is a positive strictly decreasing sequence and limk (k + 1) bk = Pn i L, 0 6= L < 1: If g (z) = i=1 ; where i are distinct and i 6= 0 for each 1 iz i. Then the following equations hold for Rbg : Sn Sn 1 (i) (Rbg ) = S i=1 D ( i L) [ E; where E = fg (0) bk gk=0 n i=1 D ( i L); n (ii) e (Rbg ) = i=1 @D ( i L), Pn (iii) For 2 = e (Rbg ), ind (Rbg ) = G ( ) ; where G = i=1 D( i L) .

GENERALIZED RHALY M ATRICES

Proof. Since g (z) =

Pn

i

i=1

1

iz

721

, we have

Rbg =

n X

i Rb

i

.

(13)

i=1

From (7), (8), (9) and Lemma 11, we obtain that f0g [ (

(Rbg ))

n X

= f0g [ =

i Rb

!

i

i=1

n [

i Rb

:

i

i=1

Since the essential spectrum of the operator T is the spectrum of (T ) coset in the Calkin algebra, i.e., e (T ) = ( (T )) and from Lemma 13, we get f0g [

e

(Rbg ) =

n [

i Rb

e

i

=

i=1

n [

@D ( i L) .

(14)

i=1

However, we know that 0 is a limit point of the right side from (14), 0 2 Therefore n [ g @D ( i L) : e (Rb ) =

e

(Rbg ). (15)

i=1

Similarly, ai Rb i aj Rb j 2 K H 2 since Rb i Rb j 2 B2 H 2 a vector space where

K H 2 and K H 2 is

K H 2 = K : K : H 2 ! H 2 is a compact linear operator : Pn Sn i Therefore, from Lemma 12, we have 2 = e for 2 = i=1 e (Rbg ) and i=1 i Rb Pn ind (Rbg ) = i=1 ind i Rb i . Since ind i Rb i = D( i L) , we get ind

i Rb

i

=

1 for

2 D ( i L) and ind

From Lemma 12, we know that index for ind (Rbg

)=

Rbg

n X i=1

i Rb

= 0 for

i

is additive. Thus, we have

D(

i L)

=

G( )

2 = D( i L). (16)

Sn )= for 2 = e (Rbg ) = i=1 @D ( i L) by using Theorem 15-(iii). From (11), if ind (Rbg 0 then 2 = D( i L) for every i. Now, we observe points 2 (Rbg ) such that ind (Rbg ) = 0. Firstly, we g indicate that 2 p (Rb ) since ind (Rbg ) = 0. From de…nition of set E, Sn E \ i=1 D ( i L) = ;, we de…ne the set E := f 2 (Rbg ) : ind (Rbg ) = 0g. Since Rbg is a lower triangular matrix, from Lemma 14, we know that the only 1 possible eigenvalues of Rbg are the diagonal elements fg (0) bk gk=1 . We can easily

722

M OHAM M AD M URSALEEN, M USTAFA YILDIRIM , AND NUH DURNA

n o1 show that g (0)bk

k=1

p

(Rbg ) , because they are diagonal elements of upper

triangular operator. Therefore, E

1

fg (0) bk gk=1 : Thus,

(Rbg ) =

n [

i=1

D (Lai ) [ E:

Remark 16. If we take bn = 1= (n + 1) in Theorem 15, we obtain [19, Theorem 3.3]. Remark 17. If we take g (z) = 1= (1 3.3] and [16, Theorem 2.2].

z) in Theorem 15, we obtain [17, Theorem

Remark 18. If we take bn = 1= (n + 1) and g (z) = 1= (1 obtain [3, Theorem 2] and [9, Theorem 4].

z) in Theorem 15, we

References [1] Akhmedov, A. M. and Ba¸sar, F., The …ne spectra of the Cesaro operator C1 over the sequence space bvp , Math. J. Okayama Univ., 50 (2007), 135-147. [2] Aleman, A. and Siskakis, A. G., An integral Operator on H p , Complex Variables, 28 (1995), 149-158. [3] Brown, A., Halmos, P. R. and Shields, A. L., Cesàro Operators, Acta Sci. Math., 26 (1965), 125-134. [4] Cima, J. A. and Petersen, K .E., Some Analytic Functions Whose Boundary Values Have Bounded Mean Oscillation, Math. Z., 147 (1976), 237-247. [5] Conway, J. B., A Course in Operator Theory, AMS, 2000. [6] Durna, N. and Yildirim, M., Topological resuls about the set of generalized Rhally matrices, Gen. Math. Notes, 17(1) (2013), 1-7. [7] Durna, N. and Yildirim, M., Generalized terraced matrices, Miskolc Math. Notes, 17(1) (2016), 201-208. [8] Garnett, J. B., Bounded analytic functions, vol. 96 of Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. [9] Gonzalez, M., The …ne spectrum of the Cesàro operator in `p (1 < p < 1) ; Arch. Math., 44 (1985), 355-358. [10] Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities, Cambridge, 1934. [11] Okutoyi, I. J., On the spectrum of C1 as an operator on bv0 , J. Austral. Math. Soc. Series A, 48 (1990), 79-86. [12] Pommerenke, C., Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv., 52 (1977), 591-602. [13] Rhaly, Jr. H. C., Terraced matrices, Bull. London Math. Soc., 21 (1989), 399-406. [14] Rhoades, B. E. and Yildirim, M., The spectra of factorable matrices on `p , Integr. Equ. Oper. Theory, 55 (2006), 111-126. [15] Siskakis, A. G., Composition Semigroups and the Cesàro Operator on H p , J. London Math. Soc., 36 (1987), 153-164. [16] Valeriyevna, G. O., Strength Properties of Safara Essential Spectrum, Dissertation, Belarusian State University at Minsk, 2010. [17] Yildirim, M., On the Spectrum of the Rhaly Operators on `p , Indian J. Pure Appl. Math., 32(2) (2001), 191-198.

GENERALIZED RHALY M ATRICES

723

[18] Young, S. W., Algebraic and Spectral Properties of Generalized Cesàro Operators, Dissertation, University of North Carolina at Chapel Hill, 2002. [19] Young, S. W., Spectral Properties of Generalized Cesàro Operators, Integr. Equ. Oper. Theory, 50 (2004), 129-146. [20] Young, S. W., Generalized Cesàro Operators and the Bergman Spaces, J. Oper. Theory, 52 (2004), 341-351. Current address : Mohammad Mursaleen: Aligarh Muslim University, 202002 Aligarh, India E-mail address : [email protected] ORCID Address: https://orcid.org/0000-0003-4128-0427 Current address : Mustafa Yildirim: Sivas Cumhuriyet University, Department of Mathematics, Faculty of Science, 58140 Sivas, Turkey E-mail address : [email protected] ORCID Address: https://orcid.org/0000-0002-8880-5457 Current address : Nuh Durna: Sivas Cumhuriyet University, Department of Mathematics, Faculty of Science, 58140 Sivas, Turkey E-mail address : [email protected] ORCID Address: https://orcid.org/0000-0001-5469-7745

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