Compactness and Berezin Symbols

COMPACTNESS AND BEREZIN SYMBOLS

I. CHALENDAR, E. FRICAIN, M. GURDAL, AND M. KARAEV

Abstract. We answer a question raised by Nordgren and Rosen-thal about the Schatten-von Neumann class membership of oper-ators in standard reproducing kernel Hilbert H spaces in terms oftheir Berezin symbols.

1. Introduction

Let Ω be a subset of a topological space X such that the boundary∂Ω is non-empty. Let H be an infinite dimensional Hilbert space offunctions defined on Ω. We say that H is a reproducing kernel Hilbertspace if the following two conditions are satisfied:

(i) for any λ ∈ Ω, the functionals f 7−→ f(λ) are continuous on H;(ii) for any λ ∈ Ω, there exists fλ ∈ H such that fλ(λ) 6= 0.

According to the Riesz representation theorem, the assumption (i) im-plies that, for any λ ∈ Ω, there exists kλ ∈ H such that

f(λ) = 〈f, kλ〉H, f ∈ H.

The function kλ is called the reproducing kernel of H at point λ. Notethat by (ii), we surely have kλ 6= 0 and we denote by kλ the normalized

reproducing kernel, that is kλ = kλ/‖kλ‖H.

Following the definition of [NR94], we say that a reproducing kernel

Hilbert space H is standard if kλ 0 (weakly) as λ → ζ, for anypoint ζ ∈ ∂Ω. In [NR94], E. Nordgren and P. Rosenthal established acharacterization of compact operators acting on such spaces in termsof the Berezin symbols of their unitary orbits. Recall that if L(H)denotes the space of linear and bounded operators on H, then theBerezin symbol T of any operator T ∈ L(H) is the function defined on

2010 Mathematics Subject Classification. Primary 47B38, 4B07; Secondary47B35.

Key words and phrases. Berezin symbols, compact operators, Schatten-von Neu-mann classes, reproducing kernel Hilbert space, model spaces.

This work was supported by the PHC Bosphore 2010-2012.1

2 I. CHALENDAR, E. FRICAIN, M. GURDAL, AND M. KARAEV

Ω by

T (λ) = 〈T kλ, kλ〉H, λ ∈ H.Then the characterization of Nordgren and Rosenthal is the follow-

ing.

Theorem 1.1. [NR94, Corollary 2.8] Let H be a standard reproducingkernel Hilbert space on Ω and let A ∈ L(H). Then A is compact if andonly if

limλ→ζ

U−1AU(λ) = 0,

for every unitary operator U on H and every point ζ in ∂Ω.

In [NR94], Nordgren and Rosenthal addressed two questions:

Question 1: What can we say about compactness in reproducingkernel Hilbert spaces which are not standard?

Question 2: Is it possible to characterize the Schatten-von Neumannclass operators in terms of their Berezin symbols?

The aim of this note is to give answers to these two questions. Wealso provide a characterization of bounded operators acting on repro-ducing kernel Hilbert spaces in terms of the Berezin symbols of theirunitary orbits. We finally discuss the particular case whereH is a back-ward shift invariant subspace KΘ associated with an inner function Θ.

2. Berezin symbols, boundedness and compactness inreproducing kernel Hilbert spaces

Regarding Theorem 1.1, it is natural to ask if one can also char-acterize the boundedness of an operator in terms of its Berezin sym-bols. Contrary to the compactness question, the analogous bounded-ness problem is trivial. In fact, note that

(2.1) U−1AU(λ) = 〈U−1AUkλ, kλ〉 = 〈AUkλUkλ〉.Hence, if A is bounded, we have

sup|U−1AU(λ)| : λ ∈ Ω, U ∈ L(H) unitary <∞.Conversely, assume that there exists λ ∈ Ω satisfying

sup|U−1AU(λ)| : U ∈ L(H) unitary <∞.Given an arbitrary vector h of norm 1 in H, there exists a unitaryoperator U on H such that Ukλ = h. According to (2.1), we get

suph∈H,‖h‖=1

|〈Ah, h〉| <∞,

COMPACTNESS AND BEREZIN SYMBOLS 3

which gives that A is bounded. Therefore, we obtain that the followingassertions are equivalent:

(i) The operator A is bounded.(ii)

sup|U−1AU(λ)| : λ ∈ Ω, U ∈ L(H) unitary <∞.

(iii) There exists λ ∈ Ω such that

sup|U−1AU(λ)| : U ∈ L(H) unitary <∞.

Note that, in the boundedness problem, the hypothesis of stan-dardness of the reproducing kernel Hilbert space H is not required.Therefore, Question 1 of Nordgren and Rosenthal appears naturally.In this direction, we will show that the hypothesis of standardness ofthe Hilbert space H in Theorem 1.1 can be highly weakened. For thispurpose, for any reproducing kernel Hilbert space H on Ω (not nec-essarily standard), denote by ∂HΩ the subset of the boundary of Ωdefined by

∂HΩ = ζ ∈ ∂Ω : kλ 0 whenever λ→ ζ.

It is clear from the definitions that H is standard if and only if ∂HΩ =∂Ω. In the case where ∂HΩ 6= ∅, one can obtain an analogue of Theo-rem 1.1. The proof will follow along the same lines and the key pointis the following result which goes back to Dixmier [Dix49].

Lemma 2.1. Let (fn)n≥1 be a weakly null sequence of unit vectorsand let (δn)n≥1 be a sequence of positive numbers. Then there exists asubsequence (fnk

)k≥1 of (fn)n≥1 and an orthonormal sequence (hk)k≥1

such that ‖fnk− hk‖ < δk for all k ≥ 1.

Theorem 2.2. Let H be a reproducing kernel Hilbert space on Ω suchthat ∂HΩ 6= ∅ and let A ∈ L(H). Then the following assertions areequivalent.

(i) A is compact;(ii) for every point ζ ∈ ∂HΩ and every unitary operator U on H, we

have

limλ→ζ

U−1AU(λ) = 0;

(iii) there exists a sequence (λn)n≥1 of points in Ω, converging to apoint ζ ∈ ∂HΩ, such that for every unitary operator U on H, wehave

limn→+∞

U−1AU(λn) = 0.


Proof. The implication (i) =⇒ (ii) follows easily from (2.1) and theimplication (ii) =⇒ (iii) is trivial. Therefore the only implication tobe proved is (iii) =⇒ (i). So, let (λn)n≥1 be a sequence of points in Ω,converging to a point ζ ∈ ∂HΩ, such that

limn→+∞

U−1AU(λn) = 0,

for every unitary operator U onH. To show that A is compact, we use acharacterization due to Anderson and Stampfli [AS71, Lemma 3] whichsays that A is compact if and only if, for any orthonormal sequence(en)n, 〈Aen, en〉H → 0, as n→ +∞. Furthermore, since 〈Aen, en〉H → 0if and only if 〈Ae2n, e2n〉H → 0 and 〈Ae2n+1, e2n+1〉H → 0, we mayassume that the orthonormal sequence is such that span(en : n ≥ 1)has infinite codimension. So let us fix such a sequence (en)n≥1. By

Lemma 2.1, there exists a subsequence (kλn`)`≥1 of (kλn)n≥1 and an

orthonormal sequence (h`)`≥1 in H such that

lim`→+∞

‖kλn`− h`‖ = 0.

By taking a subsequence if necessary, we may suppose that the or-thogonal complement of the subspace generated by (hn)n≥1 is infinitedimensional. Since both sequences (hn)n≥1 and (en)n≥1 generate sub-spaces of infinite codimension, we can construct a unitary operator Uon H such that Uh` = e`, ` ≥ 1. If follows that

|〈Ae`, e`〉| = |〈AUh`, Uh`〉|≤ |〈AU(h` − kλn`

), Uh`〉|+ |〈AUkλn`, Ukλn`

〉|+

|〈AUkλn`, U(h` − kλn`

)〉|

≤ |U−1AU(λn`)|+ 2‖A‖‖kλn`

− h`‖,

and thus (〈Ae`, e`〉)`≥1 converges to 0, as `→ +∞.

Note that we give at the end of the paper an example showing thatthe assumption ∂HΩ 6= ∅ is indeed essential for our results.

3. Berezin symbols and ideals of compacts of operators

Let T ∈ L(H) and let |T | = (T ∗T )1/2 be its modulus. Then it iswell known that if T is compact, then |T | is also compact and hence itsspectrum is given by a sequence tending to zero (if it is infinite). Letsn = sn(T ), n ≥ 1, be the eigenvalues of |T | arranged decreasingly andcounted with multiplicities. The numbers sn are called the singularnumbers of T . Now let ω = (ωn)n≥1 be a nonincreasing sequence of


positive real numbers and let 1 ≤ p <∞. Then we define the class Spω

bySpω = T ∈ L(H) : T compact and (sn)n≥1 ∈ `p(ω)

and let

‖T‖Spω

=

(∑n≥1

spnωn

)1/p

.

It is an exercise to check that Spω is a two-sided closed ideal of L(H).

Moreover, we have

(3.1) s1(T ) = ‖T‖ ≤ ω−1/p1 ‖T‖Sp

ω.

In the particular case where ωn = 1, for any n ≥ 1, then Spω corresponds

to the so-called Schatten-von Neumann classes Sp and if p = 1 andωn = 1/n, n ≥ 1, then S1

ω correspond to the Matsaev class.In the sequel we use the following characterization of membership to

Spω. This result (precisely the equivalence of (i) and (ii)) is a particular

case of a more general result, true for any symmetrically normed ideal(see for instance [GK69, Theorem III.4.2.]). However, for the sakeof completeness, we give a direct proof which follows along the samelines as the classical case corresponding to the Schatten-von Neumannclasses (see [Zhu07, Theorem 1.27]).

Lemma 3.1. Let T be a compact operator on a Hilbert space H, 1 ≤p < +∞ and let ω = (ωn)n≥1 be a decreasing sequence of positive realnumbers. The following assertions are equivalent:

(i) The operator T is in Spω;

(ii) For all orthonormal sequences (en)n≥1 in H, we have∑n≥1

|〈Ten, en〉|p ωn < +∞;

(iii) For all orthonormal sequences (en)n≥1 in H with infinite codi-mension, we have∑

n≥1

|〈Ten, en〉|p ωn < +∞.

Moreover, if

Mp(T ) = sup

(∑n≥1

|〈Ten, en〉|p ωn

)1/p

: (en)n≥1 orthonormal sequence

,

then

(3.2) 4−1p min(1, 2

1p− 1

2 )Mp(T ) ≤ ‖T‖Spω≤ 21− 1

p max(1, 21p− 1

2 )Mp(T ).


Proof. The equivalence between (ii) and (iii) follows from the obser-vation that the sequence (〈Ten, en〉)n≥1 is in `p(ω) if and only if bothsequences (〈Te2n, e2n〉)n≥1 and (〈Te2n+1, e2n+1〉)n≥1 are in `p(ω).

Now let us prove the equivalence between (i) and (ii) and the esti-mates (3.2). First assume that T is self-adjoint. Then there exists anorthonormal set (σn)n≥1 of H such that, for all x ∈ H, we have

Tx =∑n≥1

tn〈x, σn〉σn,

where (tn)n≥1 is the sequence of eigenvalues of T . We can of courseassume that the sequence (tn)n≥1 is arranged so that its modulus isdecreasing to 0. Then we have sn(T ) = |tn| and

‖T‖pSp

ω=∑n≥1

|tn|pωn.

Since tn = 〈Tσn, σn〉, we get

‖T‖pSp

ω=∑n≥1

|〈Tσn, σn〉|pωn,

whence

(3.3) ‖T‖Spω≤Mp(T ).

On the other hand, for any orthonormal sequence (en)n≥1 in H, wehave:

〈Tek, ek〉 =

⟨∑n≥1

tn〈ek, σn〉σn, ek

⟩=

∑n≥1

tn|〈ek, σn〉|2.

Let q be the conjugate index of p (q = ∞ if p = 1). Since by Parse-val’s inequality

∑n≥1 |〈ek, σn〉|2 ≤ ‖ek‖2 = 1, we obtain using Holder’s

inequality,

|〈Tek, ek〉| ≤∑n≥1

|tn||〈ek, σn〉|2/p|〈ek, σn〉|2/q

≤

(∑n≥1

|tn|p|〈ek, σn〉|2)1/p

.


Therefore∑k≥1

|〈Tek, ek〉|pωk ≤∑k≥1

ωk∑n≥1

|tn|p|〈ek, σn〉|2

=∑k≥1

ωk∑n≥k+1

|tn|p|〈ek, σn〉|2 +∑k≥1

ωk∑n≤k

|tn|p|〈ek, σn〉|2

=∑k≥1

ωk∑n≥k+1

|tn|p|〈ek, σn〉|2 +∑n≥1

|tn|p∑k≥n

ωk|〈ek, σn〉|2.

Now using the decreasing of both sequences (|tn|)n≥1 and (ωn)n≥1, wededuce∑k≥1

|〈Tek, ek〉|pωk ≤∑k≥1

ωk|tk|p∑n≥k+1

|〈ek, σn〉|2 +∑n≥1

|tn|pωn∑k≥n

|〈ek, σn〉|2

≤ 2∑n≥1

|tn|pωn,

that is

(3.4) Mp(T ) ≤ 21/p‖T‖Spω.

This proves the desired result when T is self-adjoint.For the general case we may write T = T1 + iT2, where T1 = T+T ∗

2

and T2 = T−T ∗

2iare compact self-adjoint operators. It is clear that T is

in Spω if and only both T1 and T2 are in Sp

ω. Moreover, observing that

|〈Ten, en〉|2 = |〈T1en, en〉|2 + |〈T2en, en〉|2,and using a standard argument of convexity, we have

min(1, 21− p2 )|〈Ten, en〉|p ≤ |〈T1en, en〉|p+|〈T2en, en〉|p ≤ max(1, 21− p

2 )|〈Ten, en〉|p.Hence,(3.5)

min(1, 21p− 1

2 )Mp(T ) ≤(Mp

p (T1) +Mpp (T2)

) 1p ≤ max(1, 2

1p− 1

2 )Mp(T ).

Now using (3.4), we have for i = 1, 2,

Mpp (Ti) ≤ 2‖Ti‖pSp

ω≤ 2‖T‖p

Spω,

which gives the left inequality in (3.2). For the right inequality, notethat the convexity of t 7−→ tp, implies that

21p−1(Mp(T1) +Mp(T2)) ≤

(Mp

p (T1) +Mpp (T2)

) 1p ,

and with (3.3) and (3.5), we get

21p−1‖T‖Sp

ω≤ 2

1p−1(‖T1‖Sp

ω+ ‖T2‖Sp

ω) ≤ max(1, 2

1p− 1

2 )Mp(T ),

which gives the desired inequality.


Remark 3.2. Let

M ]p(T ) = sup

(∑n≥1

|〈Ten, en〉|p ωn

)1/p

where the supremum is taken over all orthonormal sequences (en)n ofinfinite codimension. Since for any othonormal sequences (en)n, wehave+∞∑n=1

|〈Ten, en〉|p ωn =+∞∑n=1

|〈Te2n, e2n〉|p ω2n++∞∑n=0

|〈Te2n+1, e2n+1〉|p ω2n+1,

we get

(3.6) M ]p(T ) ≤Mp(T ) ≤ 21/pM ]

p(T ).

We then have the following characterization of membership to Spω in

terms of Berezin symbols.

Theorem 3.3. Let H be a reproducing kernel Hilbert space on Ω suchthat ∂HΩ 6= ∅, A be a compact operator on H, ω = (ωn)n≥1 be adecreasing sequence of positive numbers and let p ≥ 1. The followingassertions are equivalent:

(i) The operator A is in Spω.

(ii) There exists a sequence (λn)n of points in Ω converging to apoint ζ in ∂HΩ such that for all unitary operator U on H, the

sequence (U−1AU(λn))n is in `p(ω).

Moreover, if

‖A‖Bpω

= sup‖(U−1AU(λn))n‖`p(ω) : U unitary operator on H,then we have

(3.7) c1(p)‖A‖Spω≤ ‖A‖Bp

ω≤ c2(p, ω)‖A‖Sp

ω,

where

c1(p) =1

2min(1, 2

12− 1

p ) and c2(p, ω) = 41p max(1, 2

12− 1

p )+2ω−1/p1

(+∞∑m=1

wmm2p

)1/p

.

Proof. Suppose first that A is in Spω and let (λn)n≥1 be any sequence

which tends to ζ ∈ ∂HΩ. By definition of ∂HΩ, the sequence of nor-malized reproducing kernels (kλn)n≥1 converges weakly to 0. According

to Lemma 2.1, there exist a subsequence (kλnm)m≥1 of (kλn)n≥1 and an

orthonormal sequence (hm)m≥1 such that

‖kλnm− hm‖ ≤

ω1/pm

m2.


Let U be an arbitrary unitary operator on H. Using (2.1), we have

|U−1AU(λnm)| = |〈AUkλnm, Ukλnm

〉|≤ |〈AU(kλnm

− hm), Ukλnm〉|+ |〈AUhm, Uhm〉|+

|〈AUhm, U(kλnm− hm)〉|

≤ |〈AUhm, Uhm〉|+ 2‖A‖‖kλnm− hm‖

≤ |〈AUhm, Uhm〉|+2‖A‖ω1/p

m

m2.

Since A ∈ Spω and since (Uhm)m≥1 is an orthonormal sequence, it

follows from Lemma 3.1 that (〈AUhm, Uhm〉)m≥1 is in `p(ω). Conse-

quently, we get that (U−1AU(λnm))m≥1 is in `p(ω). Moreover, using(3.1) and (3.2), we have

‖(U−1AU(λnm))m‖`p(ω) ≤ 41p max(1, 2

12− 1

p )‖A‖Spω

+

2ω−1/p1 ‖A‖Sp

ω

(∑m≥1

wmm2p

)1/p

= c2(p, ω)‖A‖Spω.

Conversely, assume that there exists a sequence (λn)n≥1 which tends

to ζ ∈ ∂HΩ such that for all unitary operator U onH, (U−1AU(λn))n ∈`p(ω). By Lemma 3.1, A ∈ Sp

ω if and only if (〈Aen, en〉)n≥1 is in `p(ω),for all orthonormal sequences (en)n≥1 with infinite codimension. Nowfix such a sequence (en)n≥1 and fix a real number r > 1. By Lemma 2.1,

there exist a subsequence (kλnm)m≥1 of (kλn)n≥1 and an orthonormal

sequence (hm)m≥1 in H such that

‖kλnm− hm‖ ≤

ω1/pm

mr.

By taking a subsequence if necessary, we may suppose that the or-thogonal complement of the subspace generated by (hm)m≥1 is infinitedimensional. Since both sequences (hn)n≥1 and (en)n≥1 generate sub-spaces of infinite codimension, we can construct a unitary operator U


on H such that Uhm = em, m ≥ 1. If follows that

|〈Aem, em〉| = |〈AUhm, Uhm〉|≤ |〈AU(hm − kλnm

), Uhm〉|+ |〈AUkλnm, Ukλnm

〉|+|〈AUkλnm

, U(hm − kλnm)〉|

≤ |U−1AU(λnm)|+ 2‖A‖‖kλnm− hm‖

≤ |U−1AU(λnm)|+ 2‖A‖ω1/pm

mr,

and thus (〈Aem, em〉)m≥1 is in `p(ω). Moreover, we have

‖(〈Aem, em〉)m‖`p(ω) ≤ ‖A‖Bpω

+ 2‖A‖

(+∞∑m=1

wmmrp

)1/p

.

Letting r → +∞ gives

‖(〈Aem, em〉)m‖`p(ω) ≤ ‖A‖Bpω.

Using (3.2) and (3.6), we get

‖A‖Spω≤ 21− 1

p max(1, 21p− 1

2 )Mp(A) ≤ 2 max(1, 21p− 1

2 )M ]p(A),

whence‖A‖Sp

ω≤ 2 max(1, 2

1p− 1

2 )‖A‖Bpω.

Since for standard reproducing kernel Hilbert spaces, we have ∂HΩ =∂Ω, we immediately have the following corollary.

Corollary 3.4. Let H be standard reproducing kernel Hilbert spaceon Ω, A be a compact operator on H, ω = (ωn)n≥1 be a decreasingsequence of positive numbers and let p ≥ 1. The following assertionsare equivalent:

(i) The operator A is in Spω.

(ii) There exists a sequence (λn)n of points in Ω converging to apoint ζ in ∂Ω such that for all unitary operator U on H, the

sequence (U−1AU(λn))n is in `p(ω).

In the particular case where ωn = 1, n ≥ 1, we also obtain thefollowing corollary which gives an answer to Question 2 of Nordgrenand Rosenthal.

Corollary 3.5. Let H be a standard reproducing kernel Hilbert space,A ∈ L(H) and let p ≥ 1. The following assertions are equivalent:

(i) The operator A is in the Schatten-von Neumann class Sp.


(ii) There exists a sequence (λn)n of points in Ω converging to apoint ζ in ∂Ω such that for all unitary operator U on H, the

sequence (U−1AU(λn))n is in `p.

Moreover, if

‖A‖Bp = sup‖(U−1AU(λn))n‖`p : U unitary operator on H,then we have

(3.8) c1(p)‖A‖Sp ≤ ‖A‖Bp ≤ c2(p)‖A‖Sp ,

where

c1(p) =1

2min(1, 2

12− 1

p ) and c2(p) = 41p max(1, 2

12− 1

p )+2

(+∞∑m=1

1

m2p

)1/p

.

Remark 3.6. Note that if T is a compact operator on a standardreproducing kernel Hilbert space H and if (λn)n is any sequence ofpoints in Ω converging to a point ζ in ∂Ω then, for all unitary operator

U on H, the sequence (U−1AU(λn))n is converging to 0. Hence thereexists a subsequence which is in `p(ω). But this subsequence dependson the unitary operator U . In fact, according to Corollary 3.4, if T isin Sp

ω, then one can find a sequence (λn)n≥1, which does not depend

on the unitary operator U , such that the sequence (U−1AU(λn))n is in`p(ω).

4. An example : the model spaces

In this section, we will study the example of closed invariant back-ward shift subspaces. We determine the corresponding subset ∂HΩand deduce then results concerning compactness and membership toSpω classes.Let H2 denote the Hardy space of the open unit disc D. Then it

is well known that H2 is a reproducing kernel Hilbert space whose re-producing kernel at the point λ in D is given by kλ(z) = (1 − λz)−1,z ∈ D. Now it is easy to see that H2 is a standard reproducing kernelHilbert space. If we are interested in (closed) subspaces of H2, there isno reason for these subspaces to be still standard (see [Kar] where thisquestion is discussed in its full generality). One of the most famoussubspaces of H2 are the closed backward shift invariant subspaces KΘ.Recall that given an inner function Θ (that is a bounded analytic func-tion in D whose radials limits are of modulus one almost everywhereon T = ∂D), we define KΘ by

KΘ = H2 ΘH2 = H2 ∩ΘH20 ,


where H20 = zH2. It is easy to show that KΘ is a reproducing kernel

Hilbert space whose reproducing kernel at point λ ∈ D is given by

kΘλ (z) =

1−Θ(λ)Θ(z)

1− λz, (z ∈ D).

In general, KΘ is not standard and we will provide, in this section, acharacterization in terms of Caratheodory set of Θ. Recall that if Θadmits the factorization

Θ(z) =∏n≥1

|λn|λn

λn − z1− λnz

exp

(−∫T

ζ + z

ζ − zdµ(ζ)

),

then its Caratheodory set EΘ [AC70] is defined by

EΘ =

ζ ∈ T :

∑n≥1

1− |λn|2

|ζ − λn|2+

∫T

dµ(τ)

|ζ − τ |2< +∞

.

Recall also that the following assertions are equivalent:

(i) ζ ∈ EΘ.(ii) For any function f ∈ KΘ, f(z) has a nontangential limit when

z → ζ nontangentially.(iii) We have

lim infz→ζ

1− |Θ(z)|2

1− |z|2< +∞.

The equivalence between (i) and (ii) appears in [AC70] while the equiv-alence of (ii) and (iii) is contained in [Sar94]. Note that Sarason treatsin fact the more general situation of de Branges-Rovnyak spaces (seealso [FM08] for an equivalence of (i) and (ii) in this more general situ-ation).

Moreover, if ζ ∈ EΘ, then f 7−→ f(ζ) is continuous on KΘ and thereexists kΘ

ζ ∈ KΘ such that

f(ζ) = 〈f, kΘζ 〉.

Note that kΘζ 6≡ 0, otherwise all functions in KΘ would have nontan-

gential limit 0 at ζ and the function 1−Θ(0)Θ obviously does not. It isalso known [Sar94] that if ζ ∈ EΘ, then kΘ

λ → kΘζ (in norm), as λ→ ζ

nontangentially.The following result characterizes the weakly null sequences of nor-

malized reproducing kernels.

Lemma 4.1. Let Θ be an inner function and let ζ ∈ T. Then thefollowing assertions are equivalent:

(i) kΘλ 0 (weakly) as λ→ ζ.


(ii) ζ ∈ T \ EΘ.(iii) We have

limz→ζ

1− |Θ(z)|2

1− |z|2= +∞.

Proof. (i) =⇒ (ii): argue by absurd and assume that kΘλ 0 (weakly)

as λ → ζ but ζ ∈ EΘ. Using the Ahern–Clark’s result then kΘλ → kΘ

ζ

(in norm), as λ → ζ nontangentially. That implies that kΘλ → kΘ

ζ

(in norm), as λ → ζ nontangentially. Therefore kΘζ = 0, but since

‖kΘζ ‖ = 1, we get a contradiction.

(ii) =⇒ (iii): Let ζ ∈ T \ EΘ. Then by Caratheodory’s theorem, wehave

lim infz→ζ

1− |Θ(z)|2

1− |z|2= +∞,

which implies obviously (iii).

(iii) =⇒ (i): Assume that

limz→ζ

1− |Θ(z)|2

1− |z|2= +∞.

Take f ∈ KΘ ∩H∞. Then we have

〈f, kΘλ 〉 = ‖kΘ

λ ‖−12 |f(λ)| ≤ ‖kΘ

λ ‖−12 ‖f‖∞.

But since ‖kΘλ ‖−1 =

(1−|λ|2

1−|Θ(λ)|2

)1/2

, we get that

limλ→ζ〈f, kΘ

λ 〉 = 0, (f ∈ H∞ ∩KΘ).

To conclude it remains to note that H∞ ∩KΘ is dense in KΘ (indeed,the reproducing kernels kΘ

λ belong to H∞ ∩KΘ).

We immediately obtain the following characterization of model spaceswhich are standard.

Theorem 4.2. Let Θ be an inner function. Then the space KΘ isstandard if and only if EΘ = ∅.

Using Theorem 4.2 and Theorem 1.1, we get the next result.

Corollary 4.3. Let Θ be an inner function such that EΘ = ∅ andlet A be an operator on KΘ. Then A is compact on KΘ if and only

U−1AU(λ)→ 0, as λ→ ζ, for any ζ ∈ T and for any unitary operatorU on KΘ.


If we use the notation of Section 2, Lemma 4.1 shows that, for anyinner function Θ, the set ∂KΘ

D, associated with Θ, is T\EΘ. Thus onecan reformulate in this context Theorem 2.2 and we get immediatelythe following.

Theorem 4.4. Let Θ be an inner function such that T \ EΘ 6= ∅. LetA ∈ L(KΘ). The following assertions are equivalent:

(i) The operator A is compact.(ii) There exists a sequence (λn)n in D converging to ζ ∈ T \ EΘ

such that for all unitary operator U ∈ L(H), ( ˜U−1AU(λn))nconverges to 0.

Of course, we have also a formulation of Theorem 3.4 in the contextof model spaces KΘ.

Note that there exist model spaces KΘ with the property that EΘ =T and then ∂KΘ

D = ∅ (a concrete example is given for instance in[AC70]). Now let A be a compact operator on such a space and as-sume that A is also positive and injective. Then, whenever λ → ζnontangentially for some ζ ∈ T and U is any unitary operator on KΘ,we have

˜U−1AU(λ) = 〈U−1AUkΘλ , k

Θλ 〉 = ‖A1/2UkΘ

λ ‖2 → ‖A1/2UkΘ

ζ ‖2 6= 0.

This shows that the assumption ∂HΩ 6= ∅ is indeed essential for theresults in the paper.

Acknowledgment: The authors wish to thank the referee for valuablecomments.

References

[AS71] J. H. Anderson and J. G. Stampfli. Commutators and compressions. IsraelJ. Math. 10 (1971), 433–441.

[AC70] P. R. Ahern and D. N. Clark. Radial limits and invariant subspaces. Amer.J. Math., 92:332–342, 1970.

[Dix49] J. Dixmier. Etude sur les varietes et les operateurs de Julia, avec quelquesapplications. Bull. Soc. Math. France, 77:11–101, 1949.

[FSW72] P. A. Fillmore, J. G. Stampfli and J. P. Williams. On the essential nu-merical range, the essential spectrum, and a problem of Halmos. Acta Sci.Math. (Szeged), 33 (1972), 179-192.

[FM08] E. Fricain and J. Mashreghi. Boundary behavior of functions in the deBranges–Rovnyak spaces. Complex Anal. Oper. Theory 2 (2008), no 1,87–97.


[GK69] I. C. Gohberg and M. G. Krein. Introduction to the theory of linear non-selfadjoint operators. Translated from the Russian by A. Feinstein. Transla-tions of Mathematical Monographs, vol. 18, 1969. American MathematicalSociety, Providence.

[Kar] M.T. Karaev. Use of reproducing kernels and Berezin symbols techniquein some questions of operator theory. Forum Math., DOI: 10.1515/FORM.2099.000 (to appear).

[NR94] E. Nordgren and P. Rosenthal. Boundary values of Berezin symbols. InNonselfadjoint operators and related topics (Beer Sheva, 1992), volume 73of Oper. Theory Adv. Appl., pages 362–368. Birkhauser, Basel, 1994.

[Sar94] D. Sarason. Sub-Hardy Hilbert spaces in the unit disk. University ofArkansas Lecture Notes in the Mathematical Sciences, 10. John Wiley& Sons Inc., New York, 1994. A Wiley-Interscience Publication.

[Zhu07] K. Zhu. Operator theory in function spaces, volume 138 of MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI,second edition, 2007.

I. Chalendar, Universite de Lyon; Universite Lyon 1; Institut CamilleJordan CNRS UMR 5208; 43, boulevard du 11 Novembre 1918, F-69622Villeurbanne

E-mail address: [email protected]

E. Fricain, Universite de Lyon; Universite Lyon 1; Institut CamilleJordan CNRS UMR 5208; 43, boulevard du 11 Novembre 1918, F-69622Villeurbanne


M. Gurdal, Suleyman Demirel University, Department of mathemat-ics, 32260 Isparta, Turkey


M. Karaev, Suleyman Demirel University, Isparta Vocational School,32260 Isparta, Turkey


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Compactness and Berezin Symbols