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

WhenX is a Hilbert space,xx^∗x, x^∗if and only ifx^∗is the function given byx^∗y y, xy∈X. Thus,VTin this case coincides with classicalWT. The algebra

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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, xy, yfor allx, y∈Xandλ∈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

In most cases, there are infinitely many semi-inner products onXsatisfying2.1; however, Lumer proved thatcoWT, the closed convex hull ofWT, is independent of the choice of

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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 byH^pβ, is the set of all formal power seriesfz _∞

n0fnzⁿwith

f^pf^p

H^p^β ∞ n0

fn^pβn^p<∞. 2.3

LetμK

n∈Kβn^p, forK⊆N∪ {0}. Thenμis aσ-finite measure andH^pβ L^pμ. So, the spaceH^pβis reflexive Banach space with the norm · H^pβ, and the dual ofH^pβis H^qβ^p/q, where 1/p1/q1 andβ^p/q {βn^p/q}6.

In the casep 2, the weighted Hardy spaces with βn 1, βn n1^−1/2, and βn n1^1/2 are classical Hardy space, Bergman space, and the Dirichlet space, respectively. The spaceH²βbecomes a Hilbert space with inner product

f, g ^∞

n0

a_nb_nβn², 2.4

wherefz

anzⁿandgz

bnzⁿare the elements ofH²β 7.

The notationf, gis to stand forgf, wheref∈H^pβandg ∈H^pβ^∗. Note that

f, g ^∞

n0

fn gnβn^p. 2.5

Forf∈H^pβandg∈H^qβ^p/q, withfz

a_nzⁿandgz

b_nzⁿ, 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

H^q^β^p/q ∞ n0

fn^pβn^pf^p

p<∞, ^∗g^p

p^∞

n0gn^qβn^pg^q

q<∞.

2.6

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So,f^∗ ∈ H^qβ^p/qand^∗g ∈H^pβ. Obviously, one can see that^∗f^∗ f for allf ∈H^pβ and ^∗g^∗ g for allg inH^pβ^∗. By a simple computation, we also have the following consequences:

aifα≥0 andf∈H^pβ, thenαf^∗α^p−1f^∗, biff ∈H^pβ,f, f^∗f^pp.

We define a semi-inner product onH^pβby g, f

: g, Ff

, 2.7

wheref, g∈H^pβandFf :f^2−pf^∗. Obviously, we havef, f f²_p. Lemma 2.1. IfT is a bounded linear operator onH^pβ, then

VT WT Tf, f^∗

:f∈H^p β

, f1 T_∗

g , g

:g ∈H^q β^p/q

, g1 ,

2.8

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

Proof. Suppose thatf∈H^pβ,g∈H^pβ^∗,fg1, andf, g1. 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| ^pβn^p α|gn| ^qβn^pand argfn gn ηsee8.

Hence,|fn| ^pα|gn|^q. But

1f^p

p fn^pβn^pα gn^qβn^pα, 2.10

and hence|fn| ^p|gn| ^q. On the other hand,

1fn gnβn^p

fngneîarg^fn ^gn βn^p eîηfnfn^p/qβn^peîη.

2.11

Therefore, eⁱârg^fⁿ^gⁿ 1, or equivalently eⁱârg^fⁿ eⁱârg^gⁿ. Hence, gn

|fn| ^p/qeⁱ^arg^fⁿ, org f^∗. Then, the unit ball ofH^pβis smooth, and so there is one and only one semi-inner product onH^pβwhich satisfy2.1. Then,VT WT 3. The last equality can be proved in a similar way as the first part of this proof, and so we omit it.

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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 onH^pβneeds not to be convex, even ifT 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 operatorTonH^pβthat may not be star shaped.

Theorem 3.1. LetT be a bounded linear operator onH^pβ. Then aVTis star shaped with respect to zero if and only if

VT Tf, f^∗

:f ∈H^p β

,f

p≤1

, 3.1

bifT is finite rank onH^pβand 0∈VT, thenVTis star shaped with respect to zero.

Proof. The proof is trivial, asTkf,kf^∗ k^pTf, f^∗, for each nonnegative real number kandf ∈H^pβ.

Example 3.2. Letβ1 1 andT be the linear operator onH^pβgiven by

Tf n

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

if0 f1 n0,

−

f0 if1

n1,

0 n >1.

3.2

Therefore,

VT Tf, f^∗

:f1, f∈H^p β

Tf

0f0^p/qe^−iθ⁰ Tf

1f1^p/qe^−iθ¹ :f1, f∈H^p β

, 3.3

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

VT

rs

r^p−2−s^p−2

cosθi

r^p−s^prs

r^p2s^p2 sinθ

:r^ps^p≤1

. 3.4

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−0.2 −0.1 0 0.1 0.2 1

0.5

0

−0.5

−1

Figure 1:VTforp3.

Now, let

αsup{Rez:z∈VT}sup rs

r^p−2−s^p−2

:r^ps^p≤1 , βsup{VT∩R}

sup

cosθ·rs

r^p−2−s^p−2

:r^ps^p≤1, r^p−s^prs

r^p2s^p2

sinθ0 .

3.5

We haveα > βunlessp2. If zrs

r^p−2−s^p−2

cosθi

r^p−s^prs

r^p2s^p2 sinθ

∈VT, 3.6

then the conjugate ofzis zsr

s^p−2−r^p−2

cosπθ i

s^p−r^psr

s^p2r^p2

sinπθ

∈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 operatorTonH^pβwith 0∈VT, such thatVTis not star shaped.

Proof. We proof this theorem by contradiction. Suppose that the spatial numerical range of each linear operator that allowed origin is star shaped. Ifa, z∈VT, then 0∈VT−a, and sotz−a ∈VT−afor 0≤ t≤ 1. Hence,tz 1−ta∈VT, and it follows thatVTis convex which is a contradiction to the previous example.

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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 onH^pβ contains all nonzero extreme points of its closure, and sinceH^pβis infinite dimensional, there is a compact operatorT onH^pβ, 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 onH^pβ. 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 ∈H^p β

,f

p≤1

. 4.1

For givenα∈VT, there is a sequencehnwithhnp1 andThn, h^∗_n → α. By reflex- ivity ofH^pβand Alaogul’s Theorem, there is a sequence{nk}^∞_k1, such thathnk → hin weak topology andh^∗_n_k → gin weak^∗topology for someh∈ballH^pβandg ∈ballH^pβ^∗.

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

f^∗

:f^∗m, x^∗ f

:fm, 4.2

respectively, onH^pβ^∗andH^pβ. Hence,

hnk, x^∗ −→ h, x^∗, h^∗_n_k, x

−→ g, x

, 4.3

ask → ∞. Then,

h_n_km−→hm, h^∗_n_km−→gm, 4.4

as k → ∞. But by definition h^∗_n_km |hnkm|^p/qeⁱ^arg^h^{n k}^m. Therefore, gm

|hm|^p/qeⁱ^arg^hmorgh^∗. On the other hand,

Thnk, h^∗_n_k

− Th, h^∗≤ Thnk, h^∗_n_k

− Th, h^∗_n_k Th, h^∗_n_k

− Th, h^∗ Thnk−h, h^∗_n_k Th,

h^∗_n_k−h^∗

≤ Thnk−hh^∗_n_k Th,

h^∗_n_k−h^∗.

4.5

SinceT is completely continuous andh_n_k → hweakly, thenThnk −h → 0, and hence Thnk, h^∗_n_k → 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 onH^pβ, such thatVTis convex. ThenVTis closed if and only if 0∈VT.

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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.

11 B. Sims, On numerical range and its application to Banach algebra, Ph.D. thesis, School of Mathematical and Physical Sciences, University of Newcastle, 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

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Algebra

Decision Sciences

Discrete Mathematics

Stochastic Analysis