Spatial Numerical Range of Operators on Weighted Hardy Spaces

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International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 812680,8pages

doi:10.1155/2011/812680

Research Article

Spatial Numerical Range of Operators on

Weighted Hardy Spaces

Abdolaziz Abdollahi and Mohammad Taghi Heydari

Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran

Correspondence should be addressed to Abdolaziz Abdollahi,[email protected]

Received 2 November 2010; Revised 29 December 2010; Accepted 24 January 2011

Academic Editor: Alexander Rosa

Copyrightq2011 A. Abdollahi and M. T. Heydari. 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.

We consider the spatial numerical range of operators on weighted Hardy spaces and give conditions for closedness of numerical range of compact operators. We also prove that the spatial numerical range of finite rank operators on weighted Hardy spaces is star shaped; though, in general, it does not need to be convex.

1. Introduction

For a bounded linear operatorTon a Hilbert spaceH, the numerical rangeWTis the image of the unit sphere ofHunder the quadratic formx → Tx, xassociated with the operator. More precisely,

WT {Tx, x:x∈ H, x1}. 1.1

LetX be a complex normed space with dual apaceX∗. The Banach algebra of all bounded linear operators is denoted byLX. For an operatorT ∈LX, the spatial numerical range

VTofT is defined by

VT {Tx, x∗:x∈X, x∗∈X∗, xx∗x, x∗1}. 1.2

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numerical range is defined by

VLX, T FT:F∈LX∗, F1FId. 1.3

The notion of numerical range or the classical field of values was first introduced by O. Toeplitz in 1918 for matrices. This concept was independently extended by G. Lumer and F. Bauer in the sixties to a bounded linear operator on an arbitrary Banach space. In 1975, Lightbourne and Martin1extended this concept by employing a class of seminorms generated by a family of supplementary projections.

In 2, Gaur and Husain have studied the spatial numerical range of elements of Banach algebras without identity. Specifically, the relationship between spatial numerical ranges, numerical ranges, and spectra has been investigated. Among other results, it has been shown that the closure of the spatial numerical range of an element of a Banach algebra without identity but with regular norm is exactly its numerical range as an element of the unitized algebra.

A complete survey on numerical ranges of operators can be found in the books by Bonsall and Duncan3,4; we refer the reader to these books for general information and background.

InSection 2, after giving some background material, we give useful formula for the spatial numerical range of operators on weighted Hardy space. InSection 3, we show that the spatial numerical range of an operator needs not to be convex, and we also prove that the spatial numerical range of finite rank operators is star shaped. Finally, inSection 4, we give conditions for closedness of numerical range of compact operators.

2. Preliminaries

LetX be a complex normed space with dual spaceX∗. The mapping·,· : X×X → Cis called a semi-inner product onXif the following properties are satisfied:

i xy, z x, z y, zfor allx, y, z∈X,

ii λx, y λx, yfor allx, y∈Xandλ∈C,

iii x, x≥0 for allx∈X,

iv|x, y|2_≤_{x, x}_{y, y}_{for all}_{x, y}_∈_X_and_λ_∈_C_.

In5, Lumer defined the concept of a semi-inner product onXand showed that every normed linear spaceX, · has at least one semi-inner product·,·, such that

x, x x2 x∈X. 2.1

In terms of a semi-inner product satisfying 2.1, the definition of usual numerical range for Hilbert space operator at once generalizes to give the definition of the numerical rangeWTfor a linear operator onX,

WT {Tx, x:x1}. 2.2

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semi-inner product satisfying2.1. In fact, Lumer showed thatcoWTdepends only on the norms of the operators.

The unit ball ofXis called smooth if for allx, withx1, there is a uniquex∗ ∈X∗, such thatx∗ 1 andx, x∗ 1. In this case, there is a unique semi-inner products onX

satisfying2.1, and thenVTcoincides with numerical rangeWTcorresponding to the unique semi-inner product satisfying2.1.

A principal result in spatial numerical range is a Theorem of Williams that givesσT⊆ VT, whereσTis the spectrum ofT. Also we havesee3

iWT⊆VT,

iicoWT coVT VLX, T,

iiisup{|λ|:λ∈WT}sup{|λ|:λ∈VT}.

It is of course trivial that every eigenvalue ofT is actually inVT.

Let 1 < p < ∞and {βn}_n be a sequence of positive numbers with β0 1. The weighted Hardy space, which is denoted byHp_β_{, is the set of all formal power series}_f_z

_∞

n0fnznwith

_fp

fp

Hp_β

∞

n0

_f_np

βnp<∞. 2.3

LetμK _n_∈_Kβnp, forK⊆N∪ {0}. Thenμis aσ-finite measure andHp_β _Lp_μ_{. So,}

the spaceHp_β_{is reflexive Banach space with the norm}_·

Hp_β, and the dual ofHpβis

Hq_βp/q_{, where 1}_/p₁_/q_{1 and}_βp/q _{_β_np/q_}₆_.

In the casep 2, the weighted Hardy spaces with βn 1, βn n1−1/2, and βn n11/2 are classical Hardy space, Bergman space, and the Dirichlet space, respectively. The spaceH2_β_{becomes a Hilbert space with inner product}

f, g

∞

n0

anbnβn2, 2.4

wherefz anznandgz bnznare the elements ofH2β 7.

The notationf, gis to stand forgf, wheref∈Hp_β_and_g _∈_Hp_β∗_{. Note that}

f, g

∞

n0

fngnβnp. 2.5

Forf∈Hp_β_and_g_∈_Hq_βp/q_{, with}_f_z _a

nznandgz

bnzn, we definef∗

and∗gbyf∗z |an|p−1sgnanznand∗gz |bn|q−1sgnbnzn, respectively, where for

a nonzero complex numberw,sgnw w/|w|and sgn0 0. Clearly,

f∗q_qf∗q

Hq_βp/q

∞

n0

_f_np

βnpfp_p<∞,

∗_gp

p

∞

n0

gnqβnpgq_q<∞.

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So,f∗ ∈ Hq_βp/q_and∗_g _∈_Hp_β_{. Obviously, one can see that}∗_f∗ _f _{for all}_f _∈_Hp_β

and ∗g∗ g for allg inHp_β∗_{. By a simple computation, we also have the following}

consequences:

aifα≥0 andf∈Hp_β_{, then}_αf∗_αp−1_f∗_,

biff ∈Hp_β_,_{f, f}∗_fp

p.

We define a semi-inner product onHp_β_by

g, f: g, Ff

, 2.7

wheref, g∈Hp_β_and_F

f :f2−pf∗. Obviously, we havef, f f2p.

Lemma 2.1. IfT is a bounded linear operator onHp_β_{, then}

VT WT Tf, f∗:f∈Hpβ, f1

T∗g, g:g ∈Hqβp/q, g1,

2.8

whereWTis the numerical range ofT with respect to the semi-inner product defined by2.7.

Proof. Suppose thatf∈Hp_β_,_g_∈_Hp_β∗_,_f_g_{1, and}_{f, g}_{1. Then}

1 f, g≤fg1. 2.9

So, equality occurs in Holder inequality, and hence there are complex numbers α and η independent ofn, such that|fn|p_β_np _α_|_g_n_|q_β_np_{and arg}_f_n_g_n _η_see₈_.

Hence,|fn|p_α_|_g_n_|q_{. But}

1fp_p fnpβnpα gnqβnpα, 2.10

and hence|fn|p_|_g_n_|q_{. On the other hand,}

1fngnβnp

fngneiargfn gn βnp

eiηfnfnp/qβnpeiη.

2.11

Therefore, eiargfngn 1, or equivalently eiargfn eiarggn. Hence, gn

|fn|p/q_eiargfn_{, or}_g _f∗_{. Then, the unit ball of}_Hp_β_{is smooth, and so there is one and}

only one semi-inner product onHp_β_{which satisfy}_2.1_{. Then,}_V_T _W_T₃_{. The last}

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3. Shape of the Spatial Numerical Range

The usual numerical range of a bounded linear operator on a Hilbert space is convex, and for every bounded linear operatorT on a normed spaceX, we know thatVLX, Tis convex. AlthoughVTneeds not to be convexsee3, B. E. Cain and H. Schneider proved that it is connected. Also in9, Kuliyev proved that the spatial numerical range of a given operator on a separable Banach space is pathwise connected.

Recall thatVTis star shaped with respect to zero if tz ∈ VTfor 0 ≤ t ≤ 1 and

z∈VT.

InTheorem 3.1, we give a necessary and suﬃcient condition for the numerical range of a bounded linear operator to be star shaped. In Example 3.2, we show that the spatial numerical range of linear operatorT onHp_β_{needs not to be convex, even if}_T _{is compact}

see also3. We also determine the shape ofVT, whenTis a finite rank operator. Finally, inTheorem 3.3, we prove that there is an operatorTonHp_β_{that may not be star shaped.}

Theorem 3.1. LetT be a bounded linear operator onHpβ. Then

aVTis star shaped with respect to zero if and only if

VT Tf, f∗:f ∈Hpβ,f_p≤1, 3.1

bifT is finite rank onHp_β_{and 0}_∈_V_T_{, then}_V_T_{is star shaped with respect to zero.}

Proof. The proof is trivial, asTkf,kf∗ kp_{Tf, f}∗_{, for each nonnegative real number}

kandf ∈Hp_β_.

Example 3.2. Letβ1 1 andT be the linear operator onHp_β_{given by}

Tfn

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

if0 f1 n0,

−f0 if1 n1,

0 n >1.

3.2

Therefore,

VT Tf, f∗:f1, f∈Hpβ

Tf0f0p/qe−iθ0Tf1f1p/qe−iθ1 :f1, f∈Hpβ,

3.3

whereθ0argf0andθ1argf1. By writing|f0|r,|f1|s, θθ1−θ0, we have

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−0.2 −0.1 0 0.1 0.2 1

0.5

0

−0.5

[image:6.600.198.403.100.308.2]

−1

Figure 1:VTforp3.

Now, let

αsup{Rez:z∈VT}suprsrp−2₋_sp−2_:_rp_sp_≤₁_,

βsup{VT∩R}

supcosθ·rsrp−2−sp−2:rpsp≤1, rp−sprsrp2sp2sinθ0.

3.5

We haveα > βunlessp2. If

zrsrp−2₋_sp−2_cos_θ_i_rp₋_sp_rs_rp2_sp2_sin_θ_∈_V_T_, _3.6

then the conjugate ofzis

zsrsp−2−rp−2cosπθ isp−rpsrsp2rp2sinπθ∈VT, 3.7

and soVT∗ VT. Thus,αwas attained at points above and below the real axis, and we have concluded thatVTis not convex unlessp2. InFigure 1, we draw the shape ofVT, forp3.

Theorem 3.3. There is an operatorTonHp_β_{with 0}_∈_V_T_{, such that}_V_T_{is not star shaped.}

Proof. We proof this theorem by contradiction. Suppose that the spatial numerical range of

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4. Compact Operators

SinceH2βis a Hilbert space, the numerical range of a compact operator onH2βis closed if and only if it contains the origin. Also the numerical range of a compact operator onHp_β

contains all nonzero extreme points of its closure, and sinceHp_β_{is infinite dimensional,}

there is a compact operatorT onHpβ, such thatVTis not closedsee10and page 103-109 of11. So, in general, the spatial numerical range of a compact operator needs not to be closed. In the following theorem, we give a closedness condition of such operators.

Theorem 4.1. LetTbe a compact operator onHpβ. IfVTis star shaped with respect to zero, then

it is closed.

Proof. SinceVTis star shaped with respect to zero, then byTheorem 3.1,

VT Tf, f∗:f ∈Hpβ,f_p≤1. 4.1

For givenα∈VT, there is a sequencehnwithhnp1 andThn, h∗n → α. By

reflex-ivity ofHpβand Alaogul’s Theorem, there is a sequence{nk}∞_k1, such thathnk → hin weak

topology andh∗_nk → gin weak∗topology for someh∈ballHp_β_and_g _∈_ball_Hp_β∗_.

Now, letm∈N. Define the bounded linear functionalsx,x∗by

xf∗:f∗m, x∗f:fm, 4.2

respectively, onHp_β∗_and_Hp_β_{. Hence,}

hnk, x∗ −→ h, x∗, h∗nk, x

−→ g, x, 4.3

ask → ∞. Then,

hnkm−→hm, h∗nkm−→gm, 4.4

as k → ∞. But by definition h∗_n_km |hnkm|p/qeiarghn km. Therefore, gm

|hm|p/qeiarghmorgh∗. On the other hand,

Thnk, h∗nk

− Th, h∗≤ Thnk, h∗nk

− Th, h∗_nk Th, h∗_nk− Th, h∗ Thnk−h, h∗nk Th,

h∗_nk−h∗

≤ Thnk−hh∗nk Th,

h∗_nk−h∗.

4.5

SinceT is completely continuous andhnk → hweakly, thenThnk −h → 0, and hence

Thnk, h∗nk → Th, h∗. So,αTh, h∗, and the proof is complete by using4.1.

Since in infinite dimensional spaces 0 is allowed in spectrum of any compact operator, then we have the following corollary.

Corollary 4.2. LetTbe a compact operator onHp_β_{, such that}_V_T_{is convex. Then}_V_T_{is closed}

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Acknowledgments

The authors would like to thank Professor Alexander Rosa, the Editor-in-Chief of the International Journal of Mathematics and Mathematical Sciences, and the referee for useful and helpful comments and suggestions.

References

1 J. H. Lightbourne, III and R. H. Martin Jr., “Projection seminorms and the field of values of linear operators,” Numerische Mathematik, vol. 24, no. 2, pp. 151–161, 1975.

2 A. K. Gaur and T. Husain, “Spatial numerical ranges of elements of Banach algebras,” International

Journal of Mathematics and Mathematical Sciences, vol. 12, no. 4, pp. 633–640, 1989.

3 F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of

Normed Algebras, vol. 2 of London Mathematical Society Lecture Note Series, Cambridge University Press,

London, UK, 1971.

4 F. F. Bonsall and J. Duncan, Numerical Ranges. II, London Mathematical Society Lecture Notes Series, no. 1, Cambridge University Press, New York, NY, USA, 1973.

5 G. Lumer, “Semi-inner-product spaces,” Transactions of the American Mathematical Society, vol. 100, pp. 29–43, 1961.

6 K. Seddighi, K. Hedayatiyan, and B. Yousefi, “Operators acting on certain Banach spaces of analytic functions,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 1, pp. 107–110, 1995.

7 A. L. Shields, “Weighted shift operators and analytic function theory,” in Topics in Operator Theory, Math. Surveys, no. 13, pp. 49–128, American Mathematical Society, Providence, RI, USA, 1974.

8 L. P. Kuptsov, “H ¨older inequality,” in Encyclopaedia of Mathematics, M. Hazewinkel, Ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

9 T. Y. Kuliyev, “Pathwise connectivity of the spatial numerical range,” Proceedings of the American

Mathematical Society, vol. 122, no. 4, pp. 1173–1174, 1994.

10 G. de Barra, J. R. Giles, and B. Sims, “On the numerical range of compact operators on Hilbert spaces,”

Journal of the London Mathematical Society, vol. 5, pp. 704–706, 1972.

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Spatial Numerical Range of Operators on Weighted Hardy Spaces

Research Article

Spatial Numerical Range of Operators on

Weighted Hardy Spaces

Abdolaziz Abdollahi and Mohammad Taghi Heydari

1. Introduction

2. Preliminaries

3. Shape of the Spatial Numerical Range

4. Compact Operators

Acknowledgments

References

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