On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices
OON SPECTRAL PROPERTIES OF COMPACT TOEPLITZOPERATORS ON BERGMAN SPACE WITHLOGARITHMICALLY DECAYING SYMBOL ANDAPPLICATIONS TO BANDED MATRICES
M. KO¨ITA, S. KUPIN, S. NABOKO, AND B. TOUR´E
Abstract.
Let L ( D ) be the space of measurable square-summable functionson the unit disk. Let L a ( D ) be the Bergman space, i.e., the (closed) subspaceof analytic functions in L ( D ). P + stays for the orthogonal projection goingfrom L ( D ) to L a ( D ). For a function ϕ ∈ L ∞ ( D ), the Toeplitz operator T ϕ : L a ( D ) → L a ( D ) is defined as T ϕ f = P + ϕf, f ∈ L a ( D ) . The main result of this article are spectral asymptotics for singular (or eigen-)values of compact Toeplitz operators with logarithmically decaying symbols,that is ϕ ( z ) = ϕ ( e iθ ) (1 + log(1 / (1 − r ))) − γ , γ > , where z = re iθ and ϕ is a continuous (or piece-wise continuous) functionon the unit circle. The result is applied to the spectral analysis of banded(including Jacobi) matrices. Introduction
The investigation of Toeplitz operators is an important topic of modern analysis.The theory of Toeplitz operators on Hardy spaces was developed extensively in 70-90’s of the last century, see monographs by Nikolski [8, 9] and B¨ottcher-Silbermann[2] for a detailed account on the topic. The study of Toeplitz operators on largerfunctional spaces ( i.e.,
Fock, Bergman spaces, etc.) is currently under progress.In this article, we are interested in spectral asymptotics of singular values ofa compact Toeplitz operator on the Bergman space with logarithmically decayingsymbol. We need several definitions to give the formulations of our results.Let D := { z : | z | < } and T := ∂ D = { z : | z | = 1 } be the unit disk andthe unit circle, respectively. As usual, we denote by L ( D ) = L ( D , dA ) the spaceof measurable square-summable functions on D with respect to the normalizedLebesgue measure dA ( z ) := dxdy/π, z ∈ D . The Bergman space is defined as a(closed) subspace of analytic functions lying in L ( D ), that is(0.1) L a ( D ) := L a ( D , dA ) = { f ∈ A ( D ) : || f || = (cid:90) D | f ( z ) | dA ( z ) < ∞} . The Riesz orthogonal projection P + : L ( D ) → L a ( D ) is given by the integraloperator(0.2) ( P + f )( z ) := (cid:90) D − z ¯ w ) f ( w ) dA ( w ) , f ∈ L ( D ) . A detailed treatment of these and other objects pertaining to Bergman spaces canbe found in Hedenmalm-Korenblum-Zhu [6], Zhu [12, Chap. 4].
Mathematics Subject Classification.
Primary: 47B35; Secondary: 30H20, 42C10.
Key words and phrases. compact Toeplitz operators, Bergman spaces, spectral asymptotics,generalized Schatten-von Neumann classes, special classes of compact operators with logarithmi-cally decaying singular values. a r X i v : . [ m a t h . SP ] J un M. KO¨ITA, S. KUPIN, S. NABOKO, AND B. TOUR´E
For a symbol ϕ ∈ L ∞ ( D ), the corresponding Toeplitz operator is defined by therelation(0.3) T ϕ f := P + ϕf, f ∈ L a ( D ) . Many analytic properties of these operators are very-well understood, see Zhu [12,Chap. 7] for a nice presentation of the subject. For instance, we have trivailly || T ϕ || ≤ || ϕ || ∞ , so the Toeplitz operator corresponding to a bounded symbol is also bounded. Thecompactness of T ϕ is related to the behavior of the symbol ϕ on the vicinity of theunit circle T . The following proposition is not difficult to prove. Proposition 0.1 ([12, Prop. 7.3]) . Let ϕ ∈ C ( ¯ D ) , the class of contionuous func-tions on the closure of D . Then, T ϕ is compact if and only if lim | z |→ − ϕ ( z ) = 0 , z ∈ D . For a positive symbol ϕ , criteria for T ϕ to be compact or to belong to Schatten-von Neumann classes S p , < p < ∞ , can be found in [12, Sect. 7.3].Subsequent results in this direction address the spectral asymptotics ( i.e., theasymptotics of eigen- and/or singular values) of Toeplitz operators with symbolsfrom some special classes.Consider a symbol ϕ of the following form(0.4) ϕ ( z ) := ϕ ( e iθ ) ϕ ( r ) = ϕ ( e iθ ) (1 − r ) γ , where ϕ ∈ L /γ ( T ) , γ >
0, and z := re iθ ∈ D . By Proposition 0.1, the Toeplitzoperator T ϕ is obviously compact. Its singular values { s n ( T ϕ ) } n form a decreasingsequence and converge to zero. An important result to come was obtained inPushnitski [10]. Theorem 0.2 ([10, Thm. 1.1]) . Consider the Toeplitz operator T ϕ with symboldefined in (0.4) . Its singular values { s n ( T ϕ ) } n have the following asymptotics lim n → + ∞ n γ s n ( T ϕ ) = Γ( γ + 1)2 γ || ϕ || L /γ ( T ) (0.5) := Γ( γ + 1)2 γ (cid:18)(cid:90) π | ϕ ( e iθ ) | /γ dθ π (cid:19) γ . Above, Γ( . ) is Euler gamma-function. The above result was obtained in [10] in slightly more general form. Definingthe counting function n ( ., T ϕ ) for the sequence of singular numbers of T ϕ as n ( s, T ϕ ) := { n : s n ( T ϕ ) > s } , s > , we can rewrite (0.5) in an equivalent mannerlim s → s /γ n ( s, T ϕ ) = Γ( γ + 1) /γ || ϕ || /γL /γ ( T ) . The proof of the above theorem is purely operator-theoretic and it uses some basicfacts on underlying Bergman space only. We also mention a recent article El-Fallah-El-Ibbaoui [4] on a closely related topic.The main result of the present article is a counterpart of Theorem 0.2 in loga-rithmic scale. Let ϕ ∈ C ( T ), the space of continuous functions on the unit circle.For a fixed γ >
0, consider(0.6) ϕ ( z ) := ϕ ( e iθ ) ϕ ( r ) , OEPLITZ OPERATORS ON BERGMAN SPACE 3 where(0.7) ϕ ( r ) := ϕ ,γ ( r ) = 1 (cid:16) − r ) (cid:17) γ , r ∈ [0 , . That is, the symbol ϕ ( z ) logarithmically tends to zero when z goes to the unit circle T . Theorem 0.3.
Let ϕ be the a symbol defined above, γ > . The following asymp-totics hold for the singular values of Toeplitz operator T ϕ (0.8) lim n → + ∞ (log( n + 1)) γ s n ( T γ ) = || ϕ || L ∞ ( T ) . Generalizations of the above theorem are given in Section 4.1.Recalling the definition of the counting function, we can rewrite (0.8) as(0.9) lim s → s /γ log n ( s, T ϕ ) = || ϕ || /γL ∞ ( T ) , where ϕ (cid:54)≡ T . We stress that despite a partial similarity, certain importantpieces of the proof of Theorem 0.3 seem to be more involved than those of Theorem0.2. Among other technical points, it relies essentially on the structure of Bergmanspace and it uses fine properties of the Riesz orthoprojector (0.2) on it.The organization of the article is rather straightforward. A fast sample of thetheory of compact operators with logarithmically decaying singular values is devel-oped in Section 1. This section also contains a result on asymptotic orthogonalityof certain operators which will be the cornerstone for the proof of Theorem 0.3. It isproved in Section 2. The asymptotic orthogonality for a specific family of Toeplitzoperators is obtained in Section 3. Section 4 gives slightly different versions theobtained results as well as their applications to the study of spectral properties ofcompact banded matrices with logarithmically decaying entries.Throughout the article, “generic” constants change from one relation to another.Significant constants are sub-indexed like C , C , etc. Points of the unit disk D areoften written as z = | z | e iθ := re iθ , r ∈ [0 , , θ ∈ [0 , π ). As usual, the unit circle T = { z = e iθ : θ ∈ [0 , π ) } is identified with the interval [0 , π ].1. Some special classes of compact operators
Basic notions and Schatten-von Neumann classes of compact opera-tors.
In the paper, we shall use a number of basic facts on compact operators, seeGohberg-Krein [5, Chap. 2] and Birman-Solomyak [1, Chap. 11].Let A be a compact operator on a Hilbert space. The class of compact operatorson the space is denoted by S ∞ . It is well-known that the spectrum σ ( A ) of aself-adjoint compact operator A consists of the closure of the set of eignevalues { λ n ( A ) } n =1 ,..., ∞ tending to zero. The singular values of a compact operator A aredefined as s n ( A ) := λ n ( A ∗ A ) / , n ≥ . The sequence of singular values { s n ( A ) } n =1 ,..., ∞ is written taking into account themultiplicities; it is positive, decreasing and s n ( A ) goes to zero as n → + ∞ . It is asimple fact that for A ∈ S ∞ and a bounded operator B , one has(1.1) s n ( BA ) ≤ || B || s n ( A ) , see [1, Chap. 11, Sect. 1] or [5, Chap. 2, Sect. 2]. Another important characteristicconnected to the sequence { s n ( A ) } n is the counting function,(1.2) n ( s, A ) := { n : s n ( A ) > s } , s > . For instance, for
A, B ∈ S ∞ and s >
0, one has(1.3) n ( s, AB ) ≤ n ( s ) + n ( s , B ) , M. KO¨ITA, S. KUPIN, S. NABOKO, AND B. TOUR´E where s = s s and s , s >
0, see [1, Chap. 11, Sect. 1] once again.The Schatten-von Neumann classes S p , < p < ∞ , are given by S p := { A ∈ S ∞ : || A || p S p = ∞ (cid:88) n =1 s n ( A ) p < ∞} , A ∈ S p ⇔ (cid:90) ∞ s p − n ( s, A ) ds < ∞ . The class S ( p = 2) is called Hilbert-Schmidt class.1.2. Some specific classes of compact operators.
In this paper, we are mainlyinterested in classes of compact operators with logarithmically decaying singularvalues. Below, we introduce the definitions and give the properties of operatorsfrom these classes. Up to certain technical aspects, the proofs of the assertionsof this subsection follow Birman-Solomyak [1, Chap. 11, Sect. 6] and they aretherefore omitted.To start with, we consider a “shifted” counting function(1.4) ˜ n ( s, A ) := n ( s, A ) + 2 , compare with (1.2). We shall see that ˜ n ( ., A ) is more convenient for our purposesfrom technical point of view. For γ >
0, setΣ γ := (cid:26) A ∈ S ∞ : s n ( A ) = O (cid:18) n + 1)) γ (cid:19)(cid:27) (1.5) = (cid:26) A ∈ S ∞ : sup n ≥ (log n ) γ s n ( A ) < + ∞ (cid:27) , Σ γ := (cid:26) A ∈ S ∞ : s n ( A ) = o (cid:18) n + 1)) γ (cid:19)(cid:27) (1.6) = (cid:26) A ∈ S ∞ : lim n → + ∞ (log n ) γ s n ( A ) = 0 (cid:27) . For a compact operator A , the following relations are equivalent in a trivial way(1.7) sup n ≥ (log( n + 1)) γ s n ( A ) < ∞ , sup s> s /γ log ˜ n ( s, A ) < ∞ . Moreover, the equality(1.8) s n ( A ) = C (log( n + 1)) γ (1 + o (1)) , n → + ∞ , is equivalent to(1.9) log ˜ n ( s, A ) = C /γ s /γ (1 + o (1)) , s → . Two remarks are in order. First, one can work similarly with the usual and the“shifted” counting functions n ( ., A ) and ˜ n ( ., A ), respectively. Second, the classesΣ γ and Σ γ can be similarly defined asΣ γ := (cid:26) A ∈ S ∞ : log ˜ n ( s, A ) = O (cid:18) s /γ (cid:19) , s > (cid:27) = (cid:26) A ∈ S ∞ : sup s> s /γ log ˜ n ( s, A ) < + ∞ (cid:27) , Σ γ := (cid:26) A ∈ S ∞ : log ˜ n ( s, A ) = o (cid:18) s /γ (cid:19) , s > (cid:27) = (cid:26) A ∈ S ∞ : lim s → s /γ log ˜ n ( s, A ) = 0 (cid:27) . OEPLITZ OPERATORS ON BERGMAN SPACE 5
Once again, let A ∈ Σ γ . The next quantities will be useful in the sequel.(1.10) ∆ γ ( A ) = lim sup s → s /γ log ˜ n ( s, A ) , δ γ ( A ) = lim inf s → s /γ log ˜ n ( s, A ) . Proposition 1.1.
Let
A, B ∈ Σ γ . Then (cid:0) ∆ γ ( A + B ) (cid:1) γ/ (1+ γ ) ≤ (cid:0) ∆ γ ( A ) (cid:1) γ/ (1+ γ ) + (cid:0) ∆ γ ( B ) (cid:1) γ/ (1+ γ ) , (1.11) | (cid:0) ∆ γ ( A ) (cid:1) γ/ (1+ γ ) − (cid:0) ∆ γ ( B ) (cid:1) γ/ (1+ γ ) | ≤ (cid:0) ∆ γ ( A − B ) (cid:1) γ/ (1+ γ ) , (1.12) | (cid:0) δ γ ( A ) (cid:1) γ/ (1+ γ ) − (cid:0) δ γ ( B ) (cid:1) γ/ (1+ γ ) | ≤ (cid:0) ∆ γ ( A − B ) (cid:1) γ/ (1+ γ ) . (1.13)The proof of the above proposition follows [1, Chap. 11, Sect. 6, Thm. 4, Cor.5]. The next proposition is its simple corollary. Proposition 1.2 (Ky-Fan-type lemma) . Let A ∈ Σ γ and B ∈ Σ γ . Then (1.14) ∆ γ ( A + B ) = ∆ γ ( A ) , δ γ ( A + B ) = δ γ ( A ) . Proposition 1.3.
We have: (1) for any p, γ > , S p ⊂ Σ γ , (2) if A, B ∈ Σ γ , then ∆ γ ( AB ) ≤ ∆ γ ( A ) + ∆ γ ( B ) , and, consequently, AB ∈ Σ γ . (3) if A ∈ Σ γ and B ∈ Σ γ , then ∆ γ ( AB ) = 0 , i.e., AB ∈ Σ γ . Asymptotical orthogonality of a set of operators.
Let A be a compactoperator and A = L (cid:88) k =1 A k . Suppose that A k ∈ S ∞ , k = 1 , . . . , L , as well. The coming abstract propositionwill prove to be a useful tool for our purposes. Up to some technical details, it is issimilar to Pushnitski [10, Theorem 2.2], and we give a fast sketch of its proof only. Proposition 1.4.
Suppose that
A, A k , k = 1 , . . . , L , are as above and the family { A k } k is asymptotically orthogonal, that is (1.15) A ∗ k A j , A k A ∗ j ∈ Σ γ , j (cid:54) = k, j, k = 1 , . . . , L. Then (1.16)∆ γ ( A ) = lim sup s → s /γ log (cid:32) L (cid:88) k =1 ˜ n ( s, A k ) (cid:33) , δ γ ( A ) = lim inf s → s /γ log (cid:32) L (cid:88) k =1 ˜ n ( s, A k ) (cid:33) . Proof.
Set H L = ⊕ Lk =1 H , and A = diag { A , . . . , A L } : H L → H L , that is A ( f , . . . , f L ) = ( A f , . . . A L f L )for arbitrary ( f , . . . f L ) ∈ H L . Consider also the embedding operator J : H L → H given by J ( f , . . . f L ) = f + · · · + f L . We have J ∗ f = ( f, . . . , f ) : H → H L . A straightforward computation shows that( JA )( JA ) ∗ f = ( A A ∗ + · · · + A L A ∗ L ) f, (1.17) ( JA )( JA ) ∗ = A ∗ A . . . A ∗ A L ... . . . ... A ∗ L A . . . A ∗ L A L , (1.18) M. KO¨ITA, S. KUPIN, S. NABOKO, AND B. TOUR´E where we used a natural block decomposition for the operator ( JA )( JA ) ∗ : H L → H L . Since the operator A is block-diagonal on H L , we see n ( s, A ) = L (cid:88) k =0 n ( s, A k ) , s > . Now, by assumption (1.15), we have substracting the diagonal parts( JA ) ∗ ( JA ) − A ∗ A ∈ Σ γ . Notice that the singular values of T ∗ T and T T ∗ coincide for any compact ope-rator T , and, in particular, ∆ γ (( JA ) ∗ ( JA )) = ∆ γ (( JA )( JA ) ∗ ). Hence, byProposition 1.2,∆ γ (( JA ) ∗ ( JA )) = ∆ γ ( A ∗ A ) = ∆ γ ( A )= lim sup s → s /γ log (cid:32) L (cid:88) k =1 n ( s, A k ) + 2 (cid:33) . Furthermore, since AA ∗ = (cid:80) Lk,j =1 A k A ∗ j , relation (1.15) and Proposition 1.2 yieldonce again AA ∗ − ( JA )( JA ) ∗ ∈ Σ γ , and ∆ γ ( AA ∗ ) = ∆ γ (( JA )( JA ) ∗ ). Putting together the above computations,we obtain∆ γ ( A ) = ∆ γ ( AA ∗ ) = ∆ γ (( JA )( JA ) ∗ ) = ∆ γ (( JA ) ∗ ( JA ))= ∆ γ ( A ) = lim sup s → s /γ log (cid:32) L (cid:88) k =1 n ( s, A k ) + 2 (cid:33) . (1.19)Passing from counting functions n ( ., A k ) to ˜ n ( ., A k ) is also obvious, and so the firstrelation in (1.16) is proved. The proof of the second relation in (1.16) is the same(with lim sup replaced by lim inf). The proposition is completed. (cid:3) Proof of the main theorem
Some notation and starting remarks.
Recall the definition of function ϕ ,see (0.6), (0.7). To begin with, we consider the simplest radial case ϕ ( e iθ ) ≡
1, sothat ϕ ( z ) := ϕ ( z ) = ϕ ( r ). Lemma 2.1.
We have (2.1) lim n → + ∞ (log( n + 1)) γ s n ( T ϕ ) = || ϕ || L ∞ ( T ) = 1 . Proof.
Since the symbol ϕ is radial, the matrix of the positive compact ope-rator T ϕ is diagonal when computed in the standard orthonormal basis of theBergman space { e n } n =0 ,... , e n ( z ) = √ n + 1 z n . Passing to polar coordinates andusing Lemma 5.1 (with g ( r ) ≡ , s n ( T ϕ ) = ( T ϕ e n , e n ) = 2( n + 1) (cid:90) r n +1 ϕ ( r ) dr = 2( n + 1) 1(2 n + 1)(log(2 n + 1)) γ (1 + o (1))= 1(log( n + 1)) γ (1 + o (1)) , n → + ∞ . (2.2) (cid:3) OEPLITZ OPERATORS ON BERGMAN SPACE 7
Now, take a natural
L > I j := [2 π ( j − /L, πj/L ) , j = 1 , . . . , L be the partition of [0 , π ) into disjoint intervals of equal length. Set χ I j to be thecharacteristic functions of the intervals I j , j = 1 , . . . , L , and, moreover(2.3) ˜ χ j := χ I j ϕ , T ˜ χ j := P + ˜ χ j = P + ( χ I j ϕ ) , see (0.7). It is clear that the operators T ˜ χ j are unitarily equivalent to T ˜ χ byrotation z (cid:55)→ e − i (2 π ( j − /L ) z, z ∈ D , and so their singular values and countingfunctions coincide, ˜ n ( s, T ˜ χ j ) = ˜ n ( s, T ˜ χ ) , s > , j = 1 , . . . , L .2.2. Proof of Theorem 0.3.
The main tools of the proofs appearing in the presentsubsection are Proposition 1.4 and Theorem 3.1 saying that the above definedoperators T ˜ χ j , j = 1 , . . . , L , form an asymptotically orthogonal family. The proofof Theorem 3.1 is given in Section 3. Recall that ϕ = ϕ ϕ , see (0.6), (0.7). Lemma 2.2.
The following claims hold true: (1)
Let ϕ := ϕ , so that ϕ ≡ on T . Then ∆ γ ( T ϕ ) = δ γ ( T ϕ ) = 1 . (2) For j = 1 , . . . , L , ∆ γ ( T ˜ χ j ) = δ γ ( T ˜ χ j ) = 1 . Proof.
Theorem 3.1 is crucial for the proof of the current lemma. Its first claim isa simple rewriting of relation (2.1) with the help of observation on the equivalenceof (1.8) and (1.9).Turning to the second claim of the lemma, we keep ϕ := ϕ (or ϕ ≡ T ).Of course, ϕ := L (cid:88) j =1 · χ j , T ϕ := T ϕ = L (cid:88) j =1 · T ˜ χ j . By Theorem 3.1, the operators T ∗ ˜ χ j T ˜ χ k , T ˜ χ j T ∗ ˜ χ k lie in Σ γ for j (cid:54) = k, j, k = 1 , . . . , L ,and so Proposition 1.4 shows that∆ γ ( T ϕ ) = lim sup s → s /γ log L (cid:88) j =1 ˜ n ( s, T ˜ χ j ) (2.4) = lim sup s → s /γ log( L ˜ n ( s, T ˜ χ )) = ∆ γ ( T ˜ χ ) . Similarly, one sees δ γ ( T ϕ ) = δ γ ( T ˜ χ ). It remains to recall that counting functions˜ n ( ., T ˜ χ j ) and ˜ n ( ., T ˜ χ ) coincide, and the lemma is proved. (cid:3) The next proposition says that the claim of Theorem 0.3 is rather simple toprove when ϕ is a step (or a “staircase”) function given by ϕ = L (cid:88) j =1 c j · χ j , where c j ∈ C , j = 1 , . . . , L . Proposition 2.3.
For a step function ϕ defined above, we have ∆ γ ( T ϕ ) = δ γ ( T ϕ ) = || ϕ || /γL ∞ ( T ) . Proof.
To ease the writing, we suppose that c := || ϕ || L ∞ ( T ) = max j =1 ,...,L | c j | ≥ c = 0 being trivial, we assume that | c | >
0. In particular,˜ n ( s, c j T ˜ χ j ) ≤ ˜ n ( s, c T ˜ χ ) , j = 1 , . . . , L, M. KO¨ITA, S. KUPIN, S. NABOKO, AND B. TOUR´E due to | c j T ˜ χ j | ≤ | c | T ˜ χ j . Applying Theorem 3.1 and Proposition 1.4 as in (2.4), wehave ∆ γ ( T ϕ ) = lim sup s → s /γ log L (cid:88) j =1 ˜ n ( s, c j T ˜ χ j ) = lim sup s → s /γ log ˜ n ( s, c T ˜ χ ) + L (cid:88) j =2 ˜ n ( s, c j T ˜ χ j ) = lim sup s → s /γ log ˜ n ( s, c T ˜ χ ) (cid:40) (cid:80) Lj =2 ˜ n ( s, c j T ˜ χ j )˜ n ( s, c T ˜ χ ) (cid:41) . The term in the figure brackets on the RHS of this relation is greater or equal toone and bounded from above, so we continue as . . . = lim sup s → s /γ log ˜ n ( s, c T ˜ χ ) = c /γ lim sup s → s /γ log ˜ n ( s, T )= c /γ ∆ γ ( T ) = c /γ . The computation for δ γ ( T ϕ ) is completely analogous, and the proof is finished. (cid:3) Lemma 2.4.
For any ϕ ∈ L ∞ ( T ) , we have (2.5) ∆ γ ( T ϕ ) ≤ || ϕ || /γL ∞ ( T ) . Proof.
Observe that inequality (2.5) is homogeneous with respect to || ϕ || /γL ∞ ( T ) ,so we can suppose || ϕ || L ∞ ( T ) = 1 without loss of generality. Setting F := T ϕ = P + ϕ : L a ( D ) → L a ( D ), we have T ∗ ϕ T ϕ = P + ¯ ϕP + ϕP + = P + ¯ ϕ ¯ ϕ P + ϕ ϕ P + ≤ P + | ϕ ϕ | P + ≤ || ϕ || L ∞ ( T ) P + | ϕ | P + ≤ F F ∗ , where we used repeatedly that X ∗ Y X ≤ || Y || X ∗ X for bounded operators X, Y ,and Y = Y ∗ . So we have that s n ( T ϕ ) = s n ( T ∗ ϕ T ϕ ) ≤ s n ( F F ∗ ), where { s n ( T ϕ ) } n and { s n ( F F ∗ ) } n are singular values of operators T ϕ and F F ∗ , respectively. Con-sequently, ˜ n ( s, T ϕ ) ≤ ˜ n ( √ s, F F ∗ ) , and we continue as∆ γ ( T ϕ ) = lim sup s → s /γ log ˜ n ( s, T ϕ ) ≤ lim sup s → s /γ log ˜ n ( √ s, F F ∗ )= lim sup s → s / (2 γ ) log ˜ n ( s, F F ∗ ) = ∆ γ ( F F ∗ ) = ∆ γ ( F ) = ∆ γ ( T ϕ ) = 1by the first claim of Lemma 2.2. The proof is finished. (cid:3) Proof of Theorem 0.3.
Let ϕ : T → C be a complex-valued continuous function.For any given (cid:15) >
0, choose a step function ˜ ϕ with the property || ϕ − ˜ ϕ || L ∞ ( T ) < (cid:15), and set ˜ ϕ := ˜ ϕ ϕ . Lemma 2.4 says that∆ γ ( T ϕ − T ˜ ϕ ) = ∆ γ ( T ϕ − ˜ ϕ ) ≤ || ϕ − ˜ ϕ || /γL ∞ ( T ) ≤ (cid:15) /γ . On the other hand, Proposition 1.1 implies | (cid:0) ∆ γ ( T ϕ ) (cid:1) γ/ (1+ γ ) − (cid:0) ∆ γ ( T ˜ ϕ ) (cid:1) γ/ (1+ γ ) | ≤ (cid:0) ∆ γ ( T ϕ − T ˜ ϕ ) (cid:1) γ/ (1+ γ ) < (cid:15) / (1+ γ ) , | (cid:0) δ γ ( T ϕ ) (cid:1) γ/ (1+ γ ) − (cid:0) δ γ ( T ˜ ϕ ) (cid:1) γ/ (1+ γ ) | ≤ (cid:0) ∆ γ ( T ϕ − T ˜ ϕ ) (cid:1) γ/ (1+ γ ) < (cid:15) / (1+ γ ) . We finish the proof passing to the limit with respect (cid:15) → (cid:50) OEPLITZ OPERATORS ON BERGMAN SPACE 9 Proof of the asymptotic orthogonality of operators T ˜ χ j and T ˜ χ k , j (cid:54) = k The purpose of this section is to prove the following theorem. Notice that in thecase ϕ ( z ) = (1 − r ) γ , γ >
0, the proof of the similar result is rather simple.
Theorem 3.1.
For j (cid:54) = k, j, k = 1 , . . . , L , the Toeplitz operators T ˜ χ j , T ˜ χ k , see (2.3) , are asymptotically orthogonal, that is T ∗ ˜ χ k T ˜ χ j , T ˜ χ k T ∗ ˜ χ j ∈ Σ γ . By default, we assume that j (cid:54) = k, j, k = 1 , . . . , L throughout this section.We present the argument for the operators T ∗ ˜ χ k T ˜ χ j , the reasoning for T ˜ χ k T ∗ ˜ χ j iscompletely similar.Since the proof of Theorem 3.1 in logarithmic case is rather involved, it is dividedinto a few steps.3.1. STEP 1. Toeplitz operators with symbols which are compactly sup-ported in D . Take a small 0 < δ < / { z : | z | ≤ − δ } , χ ( z ) = (cid:26) , | z | ≤ − δ , | z | > − δ , χ ( z ) = 1 − χ ( z ) , z ∈ D . Then, write T ˜ χ j as T ˜ χ j = T χ ˜ χ j + T χ ˜ χ j . Since the supp ( χ ˜ χ j ) is compact in D ,the singular values { s n ( T χ ˜ χ j ) } n decay exponentially, that is, there is a constant C = C ( δ ) > s n ( T χ ˜ χ j ) ≤ C (1 − δ ) n , see Zhu [12, Chap. 7]. This bound is almost trivial and it follows from the estimateof the quadratic form of the operator ( T χ ˜ χ j with the help of explicit expression(0.2) for the projection P + . We have T ∗ ˜ χ k T ˜ χ j = T ∗ χ ˜ χ k T χ ˜ χ j + T ∗ χ ˜ χ k T χ ˜ χ j + T ∗ χ ˜ χ k T χ ˜ χ j + T ∗ χ ˜ χ k T χ ˜ χ j . All operators T ˜ χ j , T χ ˜ χ j , T χ ˜ χ j are bounded and so the singular values of the firstthree operator products on the RHS of the above relation decay exponentially aswell. That is, proving that T ∗ ˜ χ k T ˜ χ j ∈ Σ γ is equivalent to saying that T ∗ χ ˜ χ k T χ ˜ χ j ∈ Σ γ .3.2. STEP 2. Products of Toeplitz operators with smooth integral kernel.
Second, let B = T ∗ χ ˜ χ k T χ ˜ χ j . Recalling point (1) of Proposition 1.3, we wish toprove B ∈ S p ⊂ Σ γ for some p > j < k − j > k + 1, and, as always, j, k = 1 , . . . , L . This means that the intervals I j and I k do not touch. Lemma 3.2.
Let j < k − or j > k + 1 . Then T ∗ χ ˜ χ k T χ ˜ χ j ∈ S , the Hilbert-Schmidt class.Proof. The proof is well-known and quite simple, see Pushnitski [10]. Set S j := { z ∈ D : z = re iθ , r ∈ (1 − δ, , θ ∈ I j } , j = 1 , . . . , L. By definition, we have χ S j = χ χ j and ˜ χ S j := χ ˜ χ j . Consequently, T ∗ ˜ χ Sk T ˜ χ Sj = P + ˜ χ S k P + ˜ χ S j P + = P + ( ˜ χ S k P + ˜ χ S j ) P + . Since || P + || = 1, we prove that the operator ( ˜ χ S k P + ˜ χ S j ) belongs to S , and thiswill give the claim of the lemma by (1.1) . Indeed, the integral operator ( ˜ χ S k P + ˜ χ S j ) can be written by (0.2) as(( ˜ χ S k P + ˜ χ S j ) f )( w ) := (cid:90) D K ( w, z ) f ( z ) dA ( z ):= (cid:90) D χ S k ( w ) ϕ ( w ) χ S j ( z ) ϕ ( z )(1 − w ¯ z ) f ( z ) dA ( z ) , f ∈ A ( D ) . Since the distance between regions S k and S j is strictly positive, the integral kernel K ( w, z ) of the operator is a bounded function. Consequently, the kernel K ( ., . ) ofthe operator lies in L ( D × D ; dA ( w ) ∧ dA ( z )) , and so the operator ˜ χ S k P + ˜ χ S j isHilbert-Schmidt, see [1, Chap. 11, Sect. 3]. The lemma is proved. (cid:3) STEP 3. The case of T ∗ ˜ χ k T ˜ χ j with neighboring intervals, k = j ± ; abound on a kernel of an integral operator. The arguments of the previoussubsections show that it remains to prove that B := T ∗ χ ˜ χ k T χ ˜ χ j ∈ Σ γ for k = j ± , j, k = 1 , . . . , L . By rotation, assume WLOG that k = 1 and j = L . Tosimplify the notation, we set δ = 2 π/L and ω := χ ˜ χ = χ S ϕ , S := (cid:26) z = re iθ : r ∈ (1 − δ, , θ ∈ (cid:18) , πL (cid:19)(cid:27) ,ω L := χ ˜ χ L = χ S L ϕ , S L := (cid:26) z = re iθ : r ∈ (1 − δ, , θ ∈ (cid:18) π ( L − L , π (cid:19)(cid:27) . Alternatively, one can write the sets S and S L as S := { z = re iθ : r ∈ (1 − δ, , θ ∈ (0 , δ ) } ,S L := { z = re iθ : r ∈ (1 − δ, , θ ∈ ( − δ, } , so that S L = ¯ S by complex conjugation. We put B = T ∗ ω T ω L , and0 ≤ D := B ∗ B = ( T ∗ ω T ω L ) ∗ ( T ∗ ω T ω L ) = T ∗ ω L T ω T ∗ ω T ω L = P + ω L P + ω P + ω P + ω L P + (3.1)Recalling inequality (1.1), we can omit two projections P + bordering the latterexpression on the left and on the right. Consequently, the singular values (oreigenvalues) s n ( D ) are bounded from above by singular values (eigenvalues) s n ( D )of the positive operator D defined as(3.2) 0 ≤ ω L P + ω P + ω P + ω L ≤ D := ω L P + ω P + ω L , where we used that X ∗ Y X ≤ || Y || X ∗ X for two bounded operators X and Y, Y ∗ = Y . Now, we are interested in upper bounds on integral kernels for operators D, D m with a natural m >
0. Let( Df )( z ) := ( D f )( z ) := (cid:90) D ,z D ( z , z ) f ( z ) dA ( z ) , ( D m f )( z ) := (cid:90) D ,z m +1 D m ( z , z m +1 ) f ( z m +1 ) dA ( z m +1 )= (cid:90) D ,z m +1 . . . (cid:90) D ,z D ( z , z ) . . . D ( z m − , z m +1 ) (cid:124) (cid:123)(cid:122) (cid:125) m f ( z m +1 ) m (cid:89) j =1 dA ( z j +1 ) , the notation for indices z j will be made clear a bit later. When needed, we shallindicate the variable of the integration as a sub-index of the integral as it is writtenabove. OEPLITZ OPERATORS ON BERGMAN SPACE 11
Proposition 3.3.
We have D ( z , z ) = (cid:90) D ,z ˜ χ S L ( z ) ˜ χ S ( z ) ˜ χ S L ( z )(1 − z ¯ z ) (1 − z ¯ z ) dA ( z ) ,D m ( z , z m +1 ) = (cid:90) D ,z · · · (cid:90) D ,z m (cid:124) (cid:123)(cid:122) (cid:125) m − (cid:81) mj =1 ˜ χ S L ( z j − ) ˜ χ S ( z j ) ˜ χ S L ( z j +1 ) (cid:81) mj =1 (1 − z j − ¯ z j ) (1 − z j ¯ z j +1 ) m (cid:89) j =2 dA ( z j ) . (3.3) Proof.
Relations (3.1), (3.2) and the form of the Riesz orthoprojector P + in theBergman space (0.2) give that, for f ∈ L a ( D ), Df ( z ) = (cid:90) D ,z ˜ χ S L ( z ) ˜ χ S ( z )(1 − z ¯ z ) (cid:18)(cid:90) D ,z ˜ χ S ( z ) ˜ χ S L ( z )(1 − z ¯ z ) f ( z ) dA ( z ) (cid:19) dA ( z )= (cid:90) D ,z D ( z , z ) f ( z ) dA ( z ) , (3.4)where, by Fubini-Tonelli theorem, D ( z , z ) = (cid:90) D ,z ˜ χ S L ( z ) ˜ χ S ( z ) ˜ χ S L ( z )(1 − z ¯ z ) (1 − z ¯ z ) dA ( z ) . (3.5)This is the first equality in (3.3); the second one is proved similarly by an elementaryinduction. (cid:3) We stress that z j ∈ S , j = 1 , . . . , m , and z j +1 ∈ S L , j = 0 , . . . , m , in thesecond relation (3.3).Before going to the proof of a coming proposition,we introduce some notation.First, set(3.6) ψ ( r ) := 1(1 + 1 / log r ) γ , / < r < . Set ˆ δ = √ δ and define slightly different domainsˆ S L := { z : | z | < , | z − | < ˆ δ, Im z < } (3.7) = { z = 1 − re iθ : 0 ≤ r < min { ˆ δ, θ } , θ ∈ (0 , π/ } , see Figure 1. It is clear that S L ⊂ ˆ S L , and so χ S L ≤ χ ˆ S L . The region ˆ S is definedfor S in the same manner.Now, it is convenient to make the polar change of variables; notice that the newvariables ( r j , θ j ) are centered at the point z = 1, and not the origin z = 0. Moreprecisely, we set z j +1 := 1 − r j +1 e − iθ j +1 ∈ ˆ S , j = 0 , . . . , m, (3.8) z j := 1 − r j e iθ j ∈ ˆ S L , j = 1 , . . . , m. We have z j +1 ∈ ˆ S and z j ∈ ˆ S L for r k ∈ (0 , √ δ ) , θ k ∈ (0 , π/ , k = 1 , . . . , m +1. Proposition 3.4.
We have the following bound on kernels D m (3.3) in terms ofthe variables introduced in (3.8) | D m ( z , z m +1 ) |≤ C (cid:90) ˆ δ ,r · · · (cid:90) ˆ δ ,r m (cid:124) (cid:123)(cid:122) (cid:125) m − χ ˆ S L ( z ) ψ ( r ) (cid:16)(cid:81) mj =2 ψ ( r j ) (cid:17) χ ˆ S L ( z m +1 ) ψ ( r m +1 ) (cid:81) mj =1 ( r j + r j +1 ) m (cid:89) j =2 dr j . (3.9) Figure 1.
Domains S , S L , and ˆ S , ˆ S L . Proof.
First, we take the modulus under the integral in the second relation (3.3).Since S ⊂ ˆ S and S L ⊂ ˆ S L , we see ˜ χ S ≤ ˜ χ ˆ S and ˜ χ S L ≤ ˜ χ ˆ S L , so we can replace˜ χ S , ˜ χ S L with ˜ χ ˆ S , ˜ χ ˆ S L under the integral. Second, to uniformize the notation, wemake the change of variables z j ∈ ˆ S (cid:55)→ ¯ z j ∈ ˆ S L . In this way, we have that z k = 1 − r k e iθ ∈ ˆ S L for all k = 2 , . . . , m .Furthermore, we get with the help of the new variable z j − z j − z j = r j − e iθ j − + r j e iθ j − r j − r j e i ( θ j − + θ j ) . Consequently, there is a C = C (ˆ δ ) > | − z j − z j | ≥ C ( r j − + r j ) . Remind that ˜ χ ˆ S L ( z ) = χ ˆ S L ( z ) ϕ ( | z | ). Concerning these factors, we notice that1 − | z k | ≤ | − z k | = r k , k = 2 , . . . , m , and so ϕ ( z k ) ≤ ψ ( r k ) . OEPLITZ OPERATORS ON BERGMAN SPACE 13
Plugging all these bounds into the second relation (3.3), we obtain | D m ( z , z m +1 ) | ≤ (cid:90) D ,z · · · (cid:90) D ,z m (cid:124) (cid:123)(cid:122) (cid:125) m − (cid:81) mj =1 ˜ χ ˆ S L ( z j − ) ˜ χ ˆ S L ( z j ) ˜ χ ˆ S L ( z j +1 ) (cid:81) mj =1 | − z j − z j | | − z j z j +1 | m (cid:89) j =2 dA ( z j ) ≤ C (cid:90) π/ ,θ (cid:90) ˆ δ ,r r dr dθ · · · (cid:90) π/ ,θ m (cid:90) ˆ δ ,r m r m dr m dθ m (cid:124) (cid:123)(cid:122) (cid:125) m − ˜ χ ˆ S L ( z ) (cid:16)(cid:81) mj =2 ψ ( r j ) (cid:17) ˜ χ ˆ S L ( z m +1 ) (cid:81) mj =1 ( r j + r j +1 ) ≤ C (cid:16) π (cid:17) m − (cid:90) ˆ δ · · · (cid:90) ˆ δ (cid:124) (cid:123)(cid:122) (cid:125) m − ˜ χ ˆ S L ( z ) (cid:16)(cid:81) mj =2 ψ ( r j ) (cid:17) ˜ χ ˆ S L ( z m +1 ) (cid:81) mj =1 ( r j + r j +1 ) m (cid:89) j =2 dr j , where we used that r j +1 / ( r j + r j +1 ) ≤ / ( r j + r j +1 ). The proposition is proved. (cid:3) STEP 4. The case of T ∗ ˜ χ k T ˜ χ j with k = j ± ; an auxiliary combinatoriallemma and the final computation. The integral from the RHS of (3.9) is com-puted on a cube ( r , . . . , r m ) ∈ C = (0 , ˆ δ ) m − . Roughly speaking, the main ideafor the calculation of this subsection is to divide the cube in standard simplexes C i ,...,i m = { ( r , . . . , r m ) ∈ C : r i ∈ (0 , ˆ δ ) , ˆ δ > r i > · · · > r i m > } , and to obtain an appropriate bound for (3.9) from above integrating on everysimplex. Here, { i j } j =2 , ,..., m is a transposition of the set { , , . . . , m } . Lemma 3.5.
Let r j ≥ , j = 1 , . . . , l , where l > is a natural number. Then ( r + r ) l − (cid:89) j =2 ( r j + r j +1 ) ( r l + r l +1 ) · ( r + r l +2 ) l − (cid:89) j = l +2 ( r j + r j +1 ) ( r l + r l +1 ) ≥ ( max i =1 ,..., l r i ) l (cid:89) j =1 r j (cid:48)(cid:48) , (3.10) where ( (cid:81) . ) (cid:48)(cid:48) means that we drop both the maximal and the minimal factors in theproduct. Suppose that we have r i > r i > . . . r i l >
0. Then the lemma says that( r + r ) l − (cid:89) j =2 ( r j + r j +1 ) ( r l + r l +1 ) · ( r + r l +2 ) l − (cid:89) j = l +2 ( r j + r j +1 ) ( r l + r l +1 ) ≥ r i l − (cid:89) j =2 r i j . Proof.
We can assume that r j > , j = 1 , . . . , l . Consider the decomposition ofthe set { r , . . . , r l } into two disjoint subsets G , G , where G = { r , r l +1 } , G = { ( r j ) j =2 ,...,l , ( r j ) j = l +2 , l } . Using the symmetry of variables r j within each group, one can reduce the generalsituation to the analysis of four cases only: 1) max j =1 ,..., l r j ∈ G , min j =1 ,..., l r j ∈ G ; 2) max j =1 ,..., l r j ∈ G , min j =1 ,..., l r j ∈ G ; 3) max j =1 ,..., l r j ∈ G , min j =1 ,..., l r j ∈ G ; 4) max j =1 ,..., l r j ∈ G , min j =1 ,..., l r j ∈ G . Case 1:
Without loss of generality, suppose r = max j =1 ,..., l r j ∈ G , r l +1 =min j =1 ,..., l r j ∈ G . We have( r + r )( r + r ) . . . ( r l + r l +1 ) · ( r + r l +2 )( r l +2 + r l +3 ) . . . ( r l + r l ) ≥ r r . . . r l · r r l +2 . . . r l = r
21 2 l (cid:89) j =2 ,j (cid:54) = l +1 r j . Case 2:
WLOG r = max j =1 ,..., l r j ∈ G , r = min j =1 ,..., l r j ∈ G . Then( r + r )( r + r ) . . . ( r l + r l +1 ) · ( r + r l +2 )( r l +2 + r l +3 ) . . . ( r l + r l ) ≥ r r . . . r l r l +1 · r r l +2 . . . r l = r
21 2 l (cid:89) j =3 r j . Case 3:
WLOG r = max j =1 ,..., l r j ∈ G , r = min j =1 ,..., l r j ∈ G . We have( r + r )( r + r ) . . . ( r l + r l +1 ) · ( r + r l +2 )( r l +2 + r l +3 ) . . . ( r l + r l ) ≥ r r r . . . r l · r l +2 r l +3 . . . r l r l +1 = r
22 2 l (cid:89) j =3 r j . Case 4:
WLOG r = max j =1 ,..., l r j ∈ G , r = min j =1 ,..., l r j ∈ G . We get( r + r )( r + r ) . . . ( r l + r l +1 ) · ( r + r l +2 )( r l +2 + r l +3 ) . . . ( r l + r l ) ≥ r r r . . . r l +1 · r r l +2 . . . r l = r
22 2 l (cid:89) j =1 ,j (cid:54) =2 , r j . The proof is finished. (cid:3)
Proposition 3.6.
The following assertions take place: (1)
We have || D m || S ≤ C (cid:90) ˆ δ · · · (cid:90) ˆ δ (cid:124) (cid:123)(cid:122) (cid:125) m (cid:81) mj =1 ψ ( r j ) ( r + r ) (cid:16)(cid:81) m − j =2 ( r j + r j +1 ) (cid:17) ( r m + r m +1 )1( r + r m ) (cid:16)(cid:81) m − j =2 m +2 ( r j + r j +1 ) (cid:17) ( r m + r m +1 ) m (cid:89) j =1 dr j . (3.11)(2) In particular, D m ∈ S (or, equivalently, D ∈ S m ) whenever γ > / (8 m ) .Proof. The proof of the proposition is lengthy, but rather elementary. For theconvenience of the reader, we first treat the case m = 1, and then go to the general m . First , let m = 1. Recall the notation introduced in Proposition 3.4. Relation(3.9) says that(3.12) | D ( z , z ) | ≤ C (cid:90) ˆ δ ˜ χ ˆ S L ( z ) ψ ( r ) ˜ χ ˆ S L ( z )( r + r )( r + r ) dr , z , z ∈ D . We continue as I := || D || S = (cid:90) D ,z (cid:90) D ,z | D ( z , z ) | dA ( z ) dA ( z )= (cid:90) ˆ S L ,z (cid:90) ˆ S L ,z | D ( z , z ) | dA ( z ) dA ( z ) . (3.13)Now, we pass to the polar coordinates translated to point z = 1 z = 1 − r e iθ , z = 1 − r e iθ OEPLITZ OPERATORS ON BERGMAN SPACE 15 exactly as we did in Proposition 3.4. Here, r j ∈ (0 , ˆ δ ) , θ j ∈ (0 , π/ , j = 1 , | D ( z , z ) | in the integal on RHS of (3.13) by the integralbound given in (3.9). Hence, the quantity in the RHS of (3.13) is written as a4-tuple integral, and it is estimated above as(3.14) I ≤ C (cid:90) ˆ δ ,r · · · (cid:90) ˆ δ ,r (cid:124) (cid:123)(cid:122) (cid:125) (cid:81) j =1 ψ ( r j ) ( r + r )( r + r ) · ( r + r )( r + r ) (cid:89) j =1 dr j . This is point (1) of the proposition for m = 1.As already mentioned, we cut the cube C = C (ˆ δ ) = (0 , ˆ δ ) in simplexes C i ...i = { ( r , r , r , r ) : ˆ δ > r i > . . . r i > } , where i k = 1 , . . . , , i k (cid:54) = i j , k (cid:54) = j, k, j = 1 , . . . ,
4. Lemma 3.5 with l = 2 saystrivially that ( r + r )( r + r ) · ( r + r )( r + r ) ≥ r i r i r i . Consequently, the upper bound for integral (3.14) on the simplex C i ...i reads as (cid:90) C i ...i (cid:81) j =1 ψ ( r j ) ( r + r )( r + r ) · ( r + r )( r + r ) (cid:89) j =1 dr j (3.15) ≤ C (cid:90) ˆ δ dr i ψ ( r i ) r i (cid:90) r i dr i ψ ( r i ) r i · · · (cid:90) r i dr i ψ ( r i ) ≤ C (cid:90) ˆ δ dr i ψ ( r i ) r i (cid:90) r i dr i ψ ( r i ) r i (cid:90) r i dr i ψ ( r i ) r i r i ≤ C (cid:90) ˆ δ dr i ψ ( r i ) r i (cid:90) r i dr i ψ ( r i ) r i r i ≤ C (cid:90) ˆ δ dr i ψ ( r i ) r i r i = C (cid:90) ˆ δ dr i ψ ( r i ) r i where we used that ψ ( r i +1 ) ≤ ψ ( r i ) since r i +1 ≤ r i and the function ψ isincreasing. The latter integral is convergent whenever 8 γ >
1, which is point (2) ofthe current proposition with m = 1. Second , we follow the lines of the proof for m = 1 in the general case, but weuse some combinatorics. We get I m := || D m || S = (cid:90) D ,z (cid:90) D ,z m +1 | D m ( z , z m +1 ) | dA ( z ) dA ( z m +1 )= (cid:90) ˆ S L ,z (cid:90) ˆ S L ,z m +1 | D m ( z , z m +1 ) | dA ( z ) dA ( z m +1 ) . Now, we bound | D ( z , z m +1 ) | from above by the product of expressions from(3.9). So, we come to a 4 m -tuple integral I m ≤ C (cid:90) ˆ δ · · · (cid:90) ˆ δ (cid:124) (cid:123)(cid:122) (cid:125) m (cid:81) mj =1 ψ ( r j ) ( r + r ) (cid:16)(cid:81) m − j =2 ( r j + r j +1 ) (cid:17) ( r m + r m +1 )1( r + r m ) (cid:16)(cid:81) m − j =2 m +2 ( r j + r j +1 ) (cid:17) ( r m + r m +1 ) m (cid:89) j =1 dr j , the computation being completely analogous to (3.14). Point (1) of the propositionis proved in the general case. We now divide the cube C = C (ˆ δ ) = (0 , ˆ δ ) m in simplexes C i ...i m = { ( r , . . . , r m ) : ˆ δ > r i > · · · > r i m > } , where i k = 1 , . . . , m, i k (cid:54) = i j , k (cid:54) = j, k, j = 1 , . . . , m . Lemma 3.5 ( l = 2 m ) gives m (cid:89) j =1 ( r j + r j +1 ) · ( r + r m +2 ) m − (cid:89) j =2 m +2 ( r j + r j +1 ) ( r m + r m +1 ) ≥ r i m − (cid:89) j =2 r i j . Consequently, the upper bound for the integral on C i ...i m is (cid:90) C i ...i m · · · ≤ C (cid:90) ˆ δ dr i ψ ( r i ) r i (cid:90) r i dr i ψ ( r i ) r i · · · (cid:90) r i m − dr i m ψ ( r i m ) . As before, we have ψ ( r i +1 ) ≤ ψ ( r i ) since ψ is increasing and 0 < r i +1 ≤ r i . Webound this integral by telescoping as in (3.15) to obtain (cid:90) C i ...i m · · · ≤ C (cid:90) ˆ δ dr i ψ ( r i ) m r i r i = C (cid:90) ˆ δ dr i ψ ( r i ) m r i . The latter integral is convergent whenever 8 mγ >
1, which is point (2) of thecurrent proposition. The proof is finished. (cid:3)
Proof of Theorem 3.1.
Indeed, fix a natural m > γ > / (8 m ). Recallthat ( T ∗ ω T ω L ) ∗ ( T ∗ ω T ω L ) ≤ D, and, by Proposition 3.6, the compact positive operator D lies in S m ⊂ Σ γ . Bymonotonicity of singular values, the operator T ∗ ω T ω L is in Σ γ as well, and thetheorem is proved. (cid:50) Applications and concluding remarks
The corollaries presented in this Section mainly follow Pushnitski [10]. Onceagain, remind the notation for functions ϕ , ϕ , ϕ , (0.6), (0.7).4.1. Different verisons of the obtained results.Corollary 4.1. (1)
Let ϕ be a function continuous on ¯ D and having the property that lim | z |→ − | ϕ ,γ ( | z | ) − ϕ ( z ) − ϕ ( e iθ ) | = 0 , z = | z | e iθ ∈ D , for some ϕ ∈ C ( T ) . Then the singular values of operator T ϕ have theasymptotics (0.8) , (0.9) . (2) In particular, let ϕ ( z ) = ϕ ( e iθ ) ϕ ,γ ( | z | ) g ( | z | ) , where g ∈ L ∞ [0 , and g (1) := lim r → − g ( r ) . Then lim n → + ∞ log( n + 1) γ s n ( T ϕ ) = (cid:0) | g (1) | || ϕ || L ∞ ( T ) (cid:1) γ , idem for the counterpart of formula (0.9) . OEPLITZ OPERATORS ON BERGMAN SPACE 17
The proof of the first claim of the corollary is similar to the reasoning of Section3 and it is omitted. The second claim is an easy consequence of the first one.The general operator-theoretic techniques developed in Pushnitski-Yafaev [11]permit one to treat the sequences of positive and negative eigen-values { λ ± n ( T ϕ ) } n of operator T ϕ with a real symbol ϕ . Recall that ϕ ± = max {± ϕ , } . For simplicity,we give the following corollary for a symbol ϕ given in (0.6), (0.7). Corollary 4.2.
Let ϕ be a real symbol as above. We have the following asymptoticsfor the sequences of positive and negative eigenvalues of operator T ϕ , respectively lim n → + ∞ log( n + 1) γ λ ± n ( T ϕ ) = || ϕ ± || γL ∞ ( T ) . Corollary 4.3.
Formulae (0.8) , (0.9) hold for (finitely) piece-wise continuous func-tions ϕ defined on T . The last corollary follows from the fact that the piece-wise continuous finctionslie in the closure of step-functions on T in L ∞ ( T )-norm.4.2. An application to finite-banded matrices.
Clearly enough, the obtainedresults allow us to handle the singular values of finite-banded matrices with loga-rithmically decaying entries. We give a couple of definitions to make this moreprecise.Let D : (cid:96) ( Z + ) → (cid:96) ( Z + ) be a compact operator. Its matrix is denoted by D := [ d i,j ] i,j =0 ,..., ∞ in the standard basis of (cid:96) ( Z + ). For a fixed N ∈ N , we say that D is a (2 N + 1)-banded matrix, if d i,j = 0 for | i − j | > N + 1. Furthermore, let { b − N , . . . , b N } be aset of 2 N + 1 complex coefficients. For a fixed γ >
0, we say that a (2 N + 1)-bandedmatrix D has logarithmically decaying entries, if d m,m + j = b j (log m ) γ (1 + o (1)) , m → + ∞ , j = − N, . . . , N.
To give the next corollary, define a specific function ϕ corresponding to coefficients { b j } j = − N,...,N as ϕ ,b ( e iθ ) := N (cid:88) j = − N b j e ijθ . Corollary 4.4.
We have the following asymptotics for the singular values of theabove (2 N + 1) -banded matrix D (4.1) lim n → + ∞ (log n ) γ s n ( D ) = || ϕ ,b || L ∞ ( T ) . The proof of the corollary follows at once from two facts. First, the singularvalues of Toeplits operator T ϕ with ϕ = ϕ ,b ϕ have asymptotics (4.1). Second,we write the operator T ϕ in the standard basis of the Bergman space L a ( D ) andwe identify it with the obtained matrix on (cid:96) ( Z + ). It is easy to see now that D − T ϕ ∈ Σ γ , and the claim of the corollary follows from Proposition 1.2.5. Appendix A: Asymptotics of a logarithmic integral
The following lemma is an easy consequence of a Watson-type lemma for Laplaceintegrals proved in Kupin-Naboko [7, Thm. 0.2, Cor. 2.4].
Lemma 5.1.
Let g ∈ L ∞ [0 , , and g (1) := lim r → − g ( r ) (cid:54) = 0 . Then, for a γ > , (cid:90) r n (1 + log 1 / (1 − r )) γ g ( r ) dr = g (1) n (log n ) γ (1 + o (1)) . Acknowledgments.
The authors would like to thank Omar El-Fallah, LeonidGolinski, Karim Kellay and Alexander Pushnitski for heplful discussions.The work is partially supported by the project ANR-18-CE40-0035.S. Naboko kindly acknowledges the support by RScF-20-11-20032 grant andKnut and Alice Wallenberg Foundation grant. A part of this research was doneduring S. Naboko’s visit to University of Bordeaux in October-November, 2019. Heis grateful to the University for the hospitality.B. Tour´e gratefully acknowledges the financial support coming from agreementsbetween University of Bordeaux and the Embassy of France in Mali, 2018.
References [1] Birman, M. Sh.; Solomjak, M. Z. Spectral theory of selfadjoint operators in Hilbert space.Translated from the 1980 Russian original by S. Khrushch¨ev and V. Peller. Mathematics andits Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987.[2] B¨ottcher, A.; Silbermann, B. Analysis of Toeplitz operators. Second edition. Springer Mono-graphs in Mathematics. Springer-Verlag, Berlin, 2006.[3] Bruneau, V.; Raikov, G. Spectral properties of harmonic Toeplitz operators and applicationsto the perturbed Krein Laplacian. Asymptot. Anal. 109 (2018), no. 1-2, 53–74.[4] El-Fallah, O.; El-Ibbaoui, M. Asymptotic estimates of the eigenvalues of Toeplitz operatorsand applications to composition operators, submitted.[5] Gohberg, I. C.; Krein, M. G. Introduction to the theory of linear nonselfadjoint operators.Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol.18 American Mathematical Society, Providence, R.I. 1969.[6] Hedenmalm, H.; Korenblum, B.; Zhu, K. Theory of Bergman spaces. Graduate Texts inMathematics, 199. Springer-Verlag, New York, 2000.[7] Kupin, S.; Naboko, S. A version of Watson lemma in logarithmic scale, submitted, arxiv.[8] Nikolski, N.. Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel,and Toeplitz. Mathematical Surveys and Monographs, 92. American Mathematical Society,Providence, RI, 2002.[9] Nikolski, N. Operators, functions, and systems: an easy reading. Vol. 2. Model operatorsand systems. Mathematical Surveys and Monographs, 93. American Mathematical Society,Providence, RI, 2002.[10] Pushnitskii, A. Spectral asymptotics for Toeplitz operators and an application to bandedmatrices. Oper. Theory Adv. Appl., Vol. 268, Birkh¨auser-Springer, 2018, 397–412.[11] Pushnitski, A.; Yafaev, D. Spectral asymptotics for compact self-adjoint Hankel operators.J. Spectr. Theory 6 (2016), no. 4, 921–953.[12] Zhu, K. Operator theory in function spaces. Second edition. Mathematical Surveys and Mono-graphs, 138. American Mathematical Society, Providence, RI, 2007.
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