Continuity of the Norm of a Composition Operator

Transcription

1 Integr. equ. oper. theory 45 (003) X/ $ /0 c 003 Birkhäuser Verlag Basel/Switzerl Integral Equations Operator Theory Continuity of the Norm of a Composition Operator David B. Pokorny Jonathan E. Shapiro Abstract. We explore the continuity of the map which, given an analytic selfmap of the disk, takes as its value the norm of the associated composition operator on the Hardy space H. We also examine the continuity the functions which assign to a self-map of the disk the Hilbert-Schmidt norm or the essential norm of the associated composition operator show these to be discontinuous. Additionally, we characterize when the norm of a composition operator is minimal. Mathematics Subject Classification (000). Primary 47-A30. Keywords. composition operator, norm.. Introduction Let D denote the unit disc of the complex plane ϕ be a holomorphic function on D with ϕ(d) D. The equation C ϕ f = f ϕ defines a composition operator on the Hardy Space H. For any such ϕ, C ϕ is a bounded operator (see [4, pg. 7]). It is known that if ϕ n ϕ weakly in H then C ϕn C ϕ weakly. If ϕ n ϕ in H then C ϕn C ϕ pointwise or strongly. These results others are found in [7]. It is also known that ϕ n ϕ in some sense does not in general imply that C ϕ C ϕn 0. However, we show that in several cases, it does imply that C ϕn C ϕ. Let ASM(D) be the set of all analytic self-maps of the disk, considered as a subset of H or H. We will denote this set ASM(D) or ASM(D) respectively, when we wish to make a distinction. Recall that the H norm is given by ϕ = We would like to thank Joel Shapiro for his helpful comments during the preparation of this work. This material is based upon work supported by the National Science Foundation under Grant No

2 35 Pokorny Shapiro IEOT sup z < ϕ(z), the H norm is given by π ϕ = ϕ(e iθ ) dθ 0 π. Let N : ASM(D) R be given by N(ϕ) = C ϕ. We will use N : ASM(D) R N : ASM(D) R to distinguish the topologies on the domain (N N N ). Stated in a different way, we are asking: For which functions ϕ ASM(D) is N or N continuous? In addition to questions of the convergence of C ϕn to C ϕ, we will consider similar questions with the operator norm replaced with either the essential norm e or the Hilbert-Schmidt norm HS. Recall that T e is the distance from T to the subspace of compact operators in the operator norm topology T HS = Te n n= where {e n } is a complete orthonormal basis. We will also often suppress the subscript write f = f for the H norm of an analytic function in D. In Section 5, we show that the well known lower bound for the norm of a composition operator C ϕ, equation (3.), is attained (when ϕ(0) 0) only when ϕ is a constant function. This appears here as Theorem 4. Chris Hammond, in [6], has recently obtained this same theorem with a different proof.. Hilbert-Schmidt Operators Let ASM(D) HS denote the set of analytic self-maps of D that induce Hilbert- Schmidt composition operators, give this set the H norm. Suppose that ϕ n ϕ are all in ASM(D) HS. Does it then follow that C ϕ n HS C ϕ HS?In general, the answer is no, as the example below demonstrates. Let [ ϕ(z) = + ( ) ] / z +z [ ( n ϕ n (z) = n+ ) / + ( n n+ z +z ) / ] where the real part of z / is always positive. We approach the Hilbert-Schmidt norms of composition operators by using the formulas (as in [4, Sec. 3.3]) C ϕ,c ψ HS = π 0 ϕ(e iθ )ψ(e iθ ) dθ π

3 Vol. 45 (003) Continuity of the Norm 353 π C ϕ dθ HS = 0 ϕ(e iθ ) π. To prove that C ϕ is Hilbert-Schmidt, we will show by direct computation that the integral π dθ 0 ϕ(e iθ ) π is finite. Since ϕ(e iθ )=ϕ(e iθ ) this will show that C ϕ HS is finite. First note that for 0 <θ<π, e iθ tan(θ/) =, +eiθ i if we set t = tan(θ/), so thus ϕ(e iθ )= + t i = + t i t ϕ(e iθ ) = + t + t ϕ(e iθ ) = =+ + t+t It can be shown using basic calculus that the integral π 0 ( + t + t. ) dθ tan(θ/) + tan(θ/) π converges. Each of the maps ϕ n has an angular derivative at because each ϕ n is in fact analytic in a disc centered at. Thus none of the C ϕn are compact thus none have finite Hilbert-Schmidt norm. n For each n, choose n+ <r n < such that C rnϕ n HS >n. Note that each r n ϕ n induces a Hilbert-Schmidt operator. Now C rnϕ n HS while r n ϕ n ϕ in H. What about ϕ such that ϕ <? Is the map N HS (ϕ) = ϕ HS continuous at such points? Suppose that ϕ < ϕ n ϕ in ASM(D) HS. There exists an r< such that for sufficiently large n, ϕ n <r.thus C ϕn C ϕ HS = C ϕ HS + C ϕn HS C ϕ,c ϕn HS C ϕn,c ϕ HS π π = 0 ϕ n + π 0 ϕ π 0 ϕ n ϕ 0 ϕ n ϕ. This in turn gives π C ϕn C ϕ ϕ(ϕ ϕ n ) π HS = 0 ( ϕ )( ϕ n ϕ) + ϕ n (ϕ n ϕ) 0 ( ϕ n )( ϕ n ϕ)

4 354 Pokorny Shapiro IEOT π ( r ) ϕ n ϕ 0 0 as n. It now follows that C ϕn HS C ϕ HS. We thus have the following Theorem. N HS is continuous at all ϕ with ϕ <. Does this theorem hold if we replace the norm on the space ASM(D) HS with the H norm? No: Let η(z) = ( ) / + i( z ) α p (z) = p z (p D). pz Then η(z) maps D to the upper half-disk (with i mapping to 0) α p is the automorphism of the disk that sends 0 to p. For any p D, C η αp HS = because η α p (z) = on a set of positive measure (an arc of the upper halfcircle in a neighborhood of i). For each n, letη n = η α i( n ). Then, as n gets large, η n maps the unit circle (at least all of it away from some neighborhood of i) to arbitrarily small segments on the real line centered at 0, so η n 0as n n. Choose r n such that n+ <r n < C rnη n HS >n.wenowhave C rnη n HS r n η n 0inH as n. Remark: It is easy to check that if ϕ ASM(D) satisfies C ϕ HS = then C rϕ HS is unbounded for r (0, ). 3. Continuity of the norm We now address the first question raised in the introduction. Is N or N continuous? As the H topology is stronger than H topology, a map ϕ at which N is continuous is a map at which N is continuous. Theorem. The map N ( thus N ) is continuous at ϕ if either ϕ(0) = 0 or ϕ is an inner map. The map N is continuous at ϕ which satisfy ϕ(d) rd for some r<. Proof. The following estimate for the norm of a composition operator is well-known (for instance in [4, Cor. 3.7]): C ϕ + ϕ(0) (3.) ϕ(0) ϕ(0) Furthermore, it is noted in [8] that the upper bound is attained if ϕ is inner. If ϕ(0) = 0, then C ϕ =ϕ n (0) 0, so the bounds given above force lim n C ϕn =.LetT L(H ) be a bounded linear operator. Define T (m) =

5 Vol. 45 (003) Continuity of the Norm 355 TK m where K m is the orthogonal projection of H onto the subspace spanned by {,z,z,...,z m }. Then T (n) T () + T (z) + + T (z m ), so if ϕ n ϕ in H ( thus in H ), one obtains C ϕn C ϕ (n) ϕ n ϕ + ϕ n ϕ + + ϕ m n ϕ m. Since ϕ k n ϕ k ϕ n ϕ ϕ k n + + ϕ k k ϕ n ϕ, we have C ϕn C ϕ (m) m ϕ n ϕ.thusϕ C ϕ (m) is a continuous map on ASM(D) for all m. As lim m C ϕ (m) = C ϕ is an increasing limit for all ϕ ASM(D), N (ϕ) is lower semi-continuous for all ϕ. In other words, if ϕ is given ε>0then there exists δ>0such that ϕ ψ δ implies C ψ C ϕ ε. Note that if ϕ n ϕ in H then ϕ n (0) ϕ(0). This can be seen using properties of subharmonic functions (see [5, Th..6.4]). For each n, g n (z) =ϕ n (z) ϕ(z) is a bounded analytic function in D g n is subharmonic so π g n = lim r 0 If, on the other h, ϕ is inner, then lim n C ϕ n lim n g n (re iθ ) dθ π g n(0). + ϕ n (0) ϕn (0) = + ϕ(0) ϕ(0) = C ϕ. Since N (ϕ) is lower semi-continuous, this gives lim C ϕ n n = C ϕ. Suppose now that ϕ n ϕ in H ϕ(d) rd for some r<. In this case, there exists an r < such that ϕ n (z) <r for sufficiently large n. We have already shown, while proving Theorem, that we can conclude that C ϕn C ϕ HS 0, so C ϕn C ϕ 0 hence C ϕn C ϕ. This theorem shows that, in several cases, C ϕn C ϕ when ϕ n ϕ in H. In general, it is not true that C ϕ C ϕn 0. It is well known that C ϕ C ϕn can stay large, even when ϕ n ϕ in H. As a simple example, consider the functions ϕ n (z) =e i π n z ϕ(z) =z. Here (C ϕ C ϕn )(z n ) = z n (e i π n z) n = z n + z n =, so C ϕn C ϕ for all n. 4. The essential norm Let ϕ n (z) =( n )z ϕ(z) =z. Since ϕ n = n <, each C ϕ n is compact hence C ϕn e = 0 for all n (see [9] for details). Using the formula for the essential norm given in [0], we have C ϕ e = lim sup w N ϕ (w)/( log w ) = (here N ϕ is the Nevanlinna counting function). Thus the map ϕ C ϕ e is not continuous everywhere on ASM(D). In fact we have the following

6 356 Pokorny Shapiro IEOT Theorem 3. If the map ϕ C ϕ e on ASM(D) is continuous at ϕ, then C ϕ is a compact operator. Proof. Suppose that C ϕ is not a compact operator. Let ϕ n = n n+ ϕ.wehave C ϕn e = 0 for all n, but C ϕ e > 0. The converse of this theorem remains open. For reference, define the maps N,e on ASM(D) N,e on ASM(D) by N,e (ϕ) =N,e (ϕ) = C ϕ e. 5. Composition operators of minimal norm Consider the upper lower bounds for the norm of a composition operator on H that appear in equation (3.). It is known precisely when C ϕ achieves this upper bound (see [8]). Namely, if ϕ(0) = 0, then C ϕ =,ifϕ(0) 0 then C ϕ = + ϕ(0) ϕ(0) if only if ϕ is inner. Suppose that ϕ is now a constant function. A straightforward calculation shows that C ϕ = ϕ(0). Does this equation characterize the constant analytic self-maps of D? The answer is yes: Theorem 4. Suppose that ϕ is an analytic self-map of D, ϕ(0) 0,ϕ is not constant. Then C ϕ > ϕ(0). Proof. Let ϕ(0) = a without loss of generality assume that a is real positive. Since non-constant analytic maps are open, there is a real b>asuch that ϕ(z) =b for some z D. Letx>0 observe that Cϕ(K 0 + xk z ) C ϕ K 0 + xk z where K t (w) = is the reproducing kernel at t D. Wehave tw C ϕ(k 0 + xk z ) K 0 + xk z = a + x ab + x b +x + x z ( +x a = a ab + x a b +x + x z Now notice that since b > a > 0, we have a ab >. If x satisfies 0 < x < ( ( z a ) ab ), then x a < x z ab x, ).

7 Vol. 45 (003) Continuity of the Norm 357 so We thus have +x + x a a < +x + x z ab b < a a +x ab + x b. +x + x z a < C ϕ(k 0 + xk z ) K 0 + xk z C ϕ. 6. Conclusion We have examined the continuity of several types of norms on composition operators with respect the symbols of those composition operators. We have proved that the map N is lower semi-continuous, indeed, continuous at all ϕ which either map 0 to 0, or are inner, N is continuous at all ϕ which map D into rd for some r<. K To our knowledge, other functions such as N L (ϕ) =sup ϕ(w) w D K w C N L3 (ϕ) =sup ϕk w w D K w have not yet been investigated in terms of continuity. (The L notation follows [].) The continuity of these maps could possibly have some relationship with that of N, since, for ϕ in a large subset of ASM(D), we have K ϕ(w) C ϕ =sup w D K w =sup C ϕ K w w D K w. This is not true for all ϕ in ASM(D), as was shown in [] discussed further in [] [3]. We conclude with the following conjecture: Conjecture. The map N (ϕ) is a continuous function on ASM(D). This would in turn demonstrate the continuity of N. We also ask if N,e N,e are continuous at ϕ if only if C ϕ is a compact operator. References [] M. Appell, P. Bourdon, J. Thrall, Norms of composition operators on Hardy spaces, Experimental Math. 5 (996), -7. [] P. Avramidou F. Jafari, On norms of composition operators on Hardy spaces, Contemp. Math. 3 (999), [3] P. Bourdon D. Retsek, Reproducing kernels norms of composition operators, Acta Sci. Math. (Szeged) 67 (00),

8 358 Pokorny Shapiro IEOT [4] C. C. Cowen B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 996. [5] P. Duren, Theory of H p Spaces, Academic Press, New York, 970. [6] C. Hammond, On the norm of a composition operator with linear fractional symbol, preprint. [7] H. J. Schwartz, Composition operators on H p, Thesis, Univ. of Toledo, 969. [8] J. H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 30(000), [9] J. H. Shapiro, Composition Operators Classical Function Theory, Springer- Verlag, New York, 993. [0] J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. () 5(987), David B. Pokorny Mathematics Department, University of California, Berkeley, CA davebrok@soda.csua.berkeley.edu URL: Jonathan E. Shapiro Mathematics Department, California Polytechnic State University, San Luis Obispo, CA jshapiro@calpoly.edu URL: Submitted: January 3, 00 Revised: March, 00 To access this journal online: