OOn polynomials in spectral projections of spinoperators
Ood Shabtai Abstract
We show that the operator norm of an arbitrary bivariate polynomial,evaluated on certain spectral projections of spin operators, converges tothe maximal value in the semiclassical limit. We contrast this limitingbehavior with that of the polynomial when evaluated on random pairs ofprojections.
Contents l ( Z ) settings . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 The L ( T ) settings . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 The pair Π T , M Eα . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Partially supported by the European Research Council Starting grant 757585 and by theIsrael Science Foundation grant 1102/20 a r X i v : . [ m a t h - ph ] F e b Introduction
Let J , J , J : C d → C d be the generators of a unitary irreducible representationof SU (2), satisfying the commutation relations[ J , J ] = iJ , [ J , J ] = iJ , [ J , J ] = iJ . Their spectrum equals the set σ d = { j, j − , ..., − j } , where 2 j + 1 = d ([1]).Let A denote the complex algebra generated by two non-commuting variables x, y which satisfy x = x, y = y . Let C [ z, w ] denote the commutative algebraof complex polynomials, and let T : A → C [ z, w ] / I (1)map f ( x, y ) to f ( z, w ), where I is the ideal generated by z − z, w − w .Let (G k ( n ) , µ k,n ) be the Grassmannian of k -dimensional subspaces of C n ,equipped with the uniform probability measure . We identify V ∈ G k ( n ) withthe orthogonal projection P : C n → V .In this paper, we will study the following questions. Questions.
Fix f ∈ A . Let (0 , ∞ ) be the indicator function of (0 , ∞ ) ⊂ R .1. What is the behavior of f (cid:0) (0 , ∞ ) ( J ) , (0 , ∞ ) ( J ) (cid:1) in the semiclassical limit d → ∞ ?2. How does it compare with f ( P, Q ) , where P, Q ∈ G (cid:98) d (cid:99) ( d ) are randomprojections?3. To what extent do the answers to questions 1, 2 change if we replace (0 , ∞ ) ( J ) , (0 , ∞ ) ( J ) ∈ G (cid:98) d (cid:99) ( d ) with spectral projections of ranks (cid:98) αd (cid:99) ,where < α ≤ ? The present paper continues an earlier one ([18]), which also examined pairsof spectral projections arising from certain specific non-commuting quantumobservables, and from spin operators in particular. However, previously werestricted our attention only to commutators (i.e., f ( x, y ) = xy − yx ), and didnot compare with random projections.The analogues of our current, more general result for spin operators (namely,Theorem 1.4) also hold for the rest of the cases considered in the previous paper(position and momentum operators, for instance). Ultimately, we suspect thatthese results are instances of a rather general phenomenon. We refer the readerto Section 5 for further details. i.e., the unique probability measure invariant under the action of the unitary group onG k ( n ). .1 Main results We consider the case α = first. Denote P ,d = (0 , ∞ ) ( J ), P ,d = (0 , ∞ ) ( J ).Let Ω n = G (cid:98) n (cid:99) ( n ) × G (cid:98) n (cid:99) ( n ), ν n = µ (cid:98) n (cid:99) ,n × µ (cid:98) n (cid:99) ,n , and I fn = (cid:90) Ω n (cid:107) f ( P, Q ) (cid:107) op dν n ( P, Q ) . The following result was essentially conjectured by D. Kazhdan.
Theorem 1.1.
Let (cid:54) = f ∈ A . There exists a constant M f > , dependingonly on f , such that max ( P,Q ) ∈ Ω n (cid:107) f ( P, Q ) (cid:107) op = lim n →∞ I fn = lim d →∞ (cid:107) f ( P ,d , P ,d ) (cid:107) op = M f .M f is the universal tight upper bound (2.9) for (cid:107) f ( P, Q ) (cid:107) op , where P, Q arearbitrary orthogonal projections on a separable Hilbert space.
Theorem 1.1 is illustrated for f ( x, y ) = xy − yx (where M f = ) in thefollowing images. The graphs are clearly dissimilar, even though the limitscoincide. Note the seemingly different convergence rates.Figure 1: (Y. Le Floch) (cid:107) [ P ,d , P ,d ] (cid:107) op as a function of d . The apparent mod 4behavior is unproven, except for the case d = 4 n + 2 (L. Polterovich).3igure 2: (cid:107) [ P, Q ] (cid:107) op for random projections of rank (cid:98) d (cid:99) , as a function of d .Theorem 1.1 is perhaps somewhat misleading, since according to our numer-ical simulations P ,d , P ,d are quite different from random pairs of projections(compare figures 1, 2 and also figures 7, 8). Moreover, it is in fact atypical, sincethe limits do not necessarily coincide when α < .Indeed, for 0 < α ≤ , consider Ω n,α = G (cid:98) αn (cid:99) ( n ) × G (cid:98) αn (cid:99) ( n ), equipped withthe product probability measure ν n,α = µ (cid:98) αn (cid:99) ,n × µ (cid:98) αn (cid:99) ,n . Denote I fn,α = (cid:90) Ω n,α (cid:107) f ( P, Q ) (cid:107) op dν n,α ( P, Q ) . Then a combination of results from [5], [2] readily implies the following.
Theorem 1.2.
Let f ∈ ker( T ) and λ α = 4 α (1 − α ) . There exists a continuous ψ f : [0 , → [0 , ∞ ) , determined by f only, such that ψ f (0) = ψ f (1) = 0 , and I fα = lim n →∞ I fn,α = max [0 ,λ α ] ψ f . (2) Note that M f = max [0 , ψ f > (for f (cid:54) = 0 ). ψ f essentially appears in [19]. Since it is continuous, so is the (increasing)function α (cid:55)→ I fα . Conclusion 1.3. If f ∈ ker( T ) , then lim α → + I fα = ψ f (0) = 0 . On the other hand,
Theorem 1.4.
Let f ∈ A . Define intervals (0 , α d ) ⊂ R containing exactly (cid:98) αd (cid:99) elements of σ d . Denote P ,α,d = (0 ,α d ) ( J ) , P ,α,d = (0 ,α d ) ( J ) . Then lim d →∞ (cid:107) f ( P ,α,d , P ,α,d ) (cid:107) op = M f . (3)4n particular, when α < , we can rigorously say that pairs of spectralprojections of spin operators are unlike random pairs of projections. Conclusion 1.5.
For every α < there exists f ∈ ker( T ) such that I fα < M f .For every (cid:54) = f ∈ ker( T ) there exists δ > such that I fα < M f for every α < δ .Thus, there is a discrepancy between (2), (3). Conclusion 1.5 is illustrated for f ( x, y ) = xy − yx , α = in the followingimages. Note that ψ f ( t ) = (cid:112) t (1 − t ), M f = in this case (see Example 1.7).Figure 3: (cid:107) (cid:2) P (cid:98) αn (cid:99) , Q (cid:98) αn (cid:99) (cid:3) (cid:107) op as a function of n for random projections. Here, α = 0 .
05. The values concentrate about 2(1 − α ) (cid:112) α (1 − α ) ≈ . (cid:107) [ P ,α,d , P ,α,d ] (cid:107) op as a function of d for α = 0 . emark 1.6. The last figure (4) provides an example of a curious, unprovenphenomenon. When α = k , our numerical simulations suggest that the graphof d (cid:55)→ (cid:107) [ P ,α,d , P ,α,d ] (cid:107) op decomposes, modulo 4, to pieces of length k . Every polynomial f ∈ A can be written uniquely as f ( x, y ) = c + f ( xy ) xy + f ( yx ) yx + f ( xy ) x + f ( yx ) y, where c ∈ C and f , f , f , f are univariate polynomials. ψ f admits a ratherconcise formula (see Theorem 2.7) in terms of f , f , f , f . For instance, when f ( x, y ) = f ( xy ) xy − f ( yx ) yx (see Example 2.10), ψ f ( t ) = | f ( t ) | (cid:112) t (1 − t ) . More specifically,
Example 1.7.
Let f ( x, y ) = ( xy ) k +1 − ( yx ) k +1 , where k ≥ . Then ψ f ( t ) = t k (cid:112) t (1 − t ) , and M f = max [0 , ψ f ( t ) = ψ f (cid:18) k + 12 k + 2 (cid:19) = 1 √ k + 2 (cid:18) − k + 2 (cid:19) k + . Thus, I fα = max [0 , α (1 − α )] ψ f = (cid:40) M f − √ k +2 ≤ α (4 α (1 − α )) k + (1 − α ) 0 < α < − √ k +2 . We further specialize and consider f ( x, y ) = xy − yx . The previous exampleshows that M f = and I fα = (cid:40)
12 12 − √ ≤ α ≤ , − α ) (cid:112) α (1 − α ) 0 < α < − √ This is illustrated in the following images.6igure 5: A simulation of randomly drawn (cid:13)(cid:13)(cid:2) P (cid:98) αn (cid:99) , Q (cid:98) αn (cid:99) (cid:3)(cid:13)(cid:13) op as a function of α for n = 1000.The corresponding image for spectral projections of spin operators is (again)very dissimilar.Figure 6: (cid:107) [ P ,α,d , P ,α,d ] (cid:107) op as a function of α , where d = 1000. The proofs of Theorems 1.2, 1.4 are independent, but both rely on the generaltheory of two projections, which we now present. Nearly all of the contents of7his section can be found in the excellent guide to the theory [2]. A few minormodifications and lemmas were added for use in subsequent parts of the work.Let H be a separable complex Hilbert space, possibly infinite dimensional.A pair of orthogonal projections P : H → V P , Q : H → V Q give rise to adecomposition of H as the orthogonal direct sum H = V ⊕ V ⊕ V ⊕ V ⊕ V ⊕ V , (4)where V = V P ∩ V Q , V = V P ∩ V ⊥ Q ,V = V ⊥ P ∩ V Q , V = V ⊥ P ∩ V ⊥ Q ,V = ( V ⊕ V ) ⊥ ∩ V P , V = ( V ⊕ V ) ⊥ ∩ V ⊥ P , so that V P = V ⊕ V ⊕ V , V ⊥ P = V ⊕ V ⊕ V . Of course, some of the summands in the decomposition (4) of H may be trivial. Remark 2.1.
Let V = V ⊕ V . Then P, Q commute on V ⊥ = (cid:76) l,k ∈{ , } V lk .Hence, unless stated otherwise, we assume throughout this section that V (cid:54) = { } . Given α lk ∈ C , where l, k ∈ { , } , we abbreviate( α , α , α , α ) = α I V ⊕ α I V ⊕ α I V ⊕ α I V : V ⊥ → V ⊥ . Clearly P | V ⊥ = (1 , , , , Q | V ⊥ = (1 , , , P | V = diag( I, P, Q is specified as follows.
Theorem 2.2 ([10]) . V (cid:54) = { } if and only if V (cid:54) = { } . If V , V (cid:54) = { } , then P = (1 , , , ⊕ P | V = (1 , , , ⊕ (cid:18) I
00 0 (cid:19) ,Q = (1 , , , ⊕ Q | V = (1 , , , ⊕ U ∗ (cid:18) I − H (cid:112) H ( I − H ) (cid:112) H ( I − H ) H (cid:19) U. Here ≤ H ≤ with ker H = ker( I − H ) = { } and U = diag ( I, R ) with R : V → V unitary. In particular, we note that
P, Q are non-commuting if and only if V (cid:54) = { } . Recall that A denotes the complex algebra generated by two non-commutingvariables x, y which satisfy the relations x = x , y = y . A basis of A as avector space is provided by the monomials1 , ( xy ) k +1 , ( yx ) k +1 , ( xy ) k x, ( yx ) k y, (5)8here k ≥
0. Thus any f ∈ A decomposes uniquely as f ( x, y ) = a + f ( xy ) xy + f ( yx ) yx + f ( xy ) x + f ( yx ) y, where f , f , f , f are complex univariate polynomials. Let r ( t ) = (cid:112) t (1 − t ),and for l, k ∈ { , } define g lk : [0 , → C by g ( t ) = a + f ( t ) + t ( f ( t ) + f ( t ) + f ( t )) ,g ( t ) = r ( t ) ( f ( t ) + f ( t )) , g ( t ) = r ( t ) ( f ( t ) + f ( t )) ,g ( t ) = a + (1 − t ) f ( t ) . Also, denote α = g (1) , α = g (0) , α = g (0) , α = g (1) . Lemma 2.3.
Let T : A → C [ z, w ] / I be the ”abelianization” map of (1). Then f ∈ ker( T ) if and only if α = α = α = α = 0 . Proof.
Write f l ( t ) = k l (cid:88) k =0 a ( l ) k t k , l = 1 , , , . A straightforward computation shows that(
T f ) ( z, w ) = a + f (0) z + f (0) w + [ f (1) + f (1) + f (1) − f (0) + f (1) − f (0)] zw. Thus we obtain the required.Next, by induction,(
P Q ) k +1 = I V ⊕ U ∗ (cid:18) ( I − H ) k +1 ( I − H ) k r ( H )0 0 (cid:19) U, ( QP ) k +1 = I V ⊕ U ∗ (cid:18) ( I − H ) k +1 I − H ) k r ( H ) 0 (cid:19) U, ( P Q ) k P = I V ⊕ U ∗ (cid:18) ( I − H ) k
00 0 (cid:19) U, ( QP ) k Q = I V ⊕ U ∗ (cid:18) ( I − H ) k +1 ( I − H ) k r ( H )( I − H ) k r ( H ) ( I − H ) k H (cid:19) U, where k ≥
0. By linearity, we obtain a precise expression for f ( P, Q ) as follows.
Conclusion 2.4 ([9]) . For every complex Hilbert space H and for any pair ofnon-commuting orthogonal projections P : H → V P , Q : H → V Q , f ( P, Q ) = a + f ( P Q ) P Q + f ( QP ) QP + f ( P Q ) P + f ( QP ) Q =( α , α , α , α ) ⊕ U ∗ (cid:18) g ( I − H ) g ( I − H ) g ( I − H ) g ( I − H ) (cid:19) U. roof. Denote f l ( t ) = (cid:80) k l k =0 a ( l ) k t k , where l = 1 , , , k l (cid:54) = 0. We computeeach of the summands f , f , f , f of f ( P, Q ) separately.The first summand is f ( P Q ) P Q = k (cid:88) k =0 a (1) k ( P Q ) k +1 =( f (1) I ) ⊕ U ∗ (cid:18) f ( I − H )( I − H ) f ( I − H ) r ( H )0 0 (cid:19) U. The second summand is f ( QP ) QP = k (cid:88) k =0 a (2) k ( QP ) k +1 =( f (1) I ) ⊕ U ∗ (cid:18) f ( I − H )( I − H ) 0 f ( I − H ) r ( H ) 0 (cid:19) U. The third summand is f ( P Q ) P = f ( P QP ) P = a (3)0 P + k − (cid:88) k =0 a (3) k +1 ( P QP ) k +1 , where k − (cid:88) k =0 a (3) k +1 ( P QP ) k +1 = (cid:16) ( f (1) − a (3)0 ) I (cid:17) ⊕ U ∗ (cid:18) (cid:80) k − k =0 a (3) k +1 ( I − H ) k +1
00 0 (cid:19) U, so f ( P Q ) P = ( f (1) , f (0) , , ⊕ U ∗ (cid:18) f ( I − H ) 00 0 (cid:19) U. Finally, f ( QP ) Q = f ( QP Q ) Q = a (4)0 Q + k − (cid:88) k =0 a (4) k +1 ( QP Q ) k +1 , so f ( QP ) Q =( f (1) , , f (0) , ⊕ U ∗ (cid:18) f ( I − H )( I − H ) f ( I − H ) r ( H ) f ( I − H ) r ( H ) f ( I − H ) H (cid:19) U. Putting everything together, we obtain the required.10 emark 2.5.
We can factor T as Π I ◦ T , where T : A → C [ z, w ] is the linearmap f ( x, y ) (cid:55)→ f ( z, w ) (unlike T , the map T is not a morphism of algebras)and Π I is the projection onto C [ z, w ] / I . Note that f ∈ ker( T ) if and only if f ( x, y ) = f ( xy ) xy − f ( yx ) yx = [ x, f ( yx ) y ] , where f is a complex univariate polynomial, and then f ( P, Q ) = [
P, f ( QP ) Q ] == (0 , , , ⊕ U ∗ (cid:18) f ( I − H ) r ( H ) − f ( I − H ) r ( H ) 0 (cid:19) U. We define (following [19]) two functions g , g ∈ C ([0 , , C ) by g = (cid:88) l,k =0 | g lk | , g = g g − g g , (6)where g , g , g , g are the functions appearing in Conclusion 2.4. Remark 2.6.
In the literature, it appears that g , g , g , g , and g , g areconsidered only as functions on σ ( I − H ) . For us, it is very useful that they aredetermined by f as continuous functions on [0 , , so as to apply by restrictionto all separable Hilbert spaces and orthogonal projections. g , g can be used to express σ ( f ( P, Q )) in terms of σ ( I − H ), since f ( P, Q ) | V has a bounded inverse if and only if g ( I − H ) does. In particular, the polynomial f ( P, Q ) f ∗ ( P, Q ) can be used to determine (cid:107) f ( P, Q ) (cid:107) op . Theorem 2.7 ([2, 19]) . Let
Λ = { ( l, k ) ∈ { , } | V lk (cid:54) = { }} . The operatornorm of f ( P, Q ) is given by (cid:107) f ( P, Q ) (cid:107) op = max (cid:26) max ( l,k ) ∈ Λ | α l,k | , max t ∈ σ ( I − H ) ψ f ( t ) (cid:27) , where ψ f ∈ C ([0 , , R ) is given by ψ f ( t ) = (cid:115) g ( t ) + (cid:112) g ( t ) − | g ( t ) | . Note that ψ f (0) = max {| α | , | α |} , ψ f (1) = max {| α | , | α |} . Assume f ∈ ker( T ), so that by Lemma 2.3, α lk = 0 for l, k = 0 ,
1, or equivalently ψ f (0) = ψ f (1) = 0. If P, Q commute, then f ( P, Q ) = 0 and max σ ( P QP ) ψ f = 0.Otherwise if P, Q do not commute, max σ ( I − H ) ψ f = max σ ( P QP ) ψ f . We usethis to reformulate Theorem 2.7 as follows.11 onclusion 2.8. Let
Λ = { ( l, k ) ∈ { , } | V lk (cid:54) = { }} as above, and f ∈ A .1. Assume f ∈ ker( T ) . Then for every separable complex Hilbert space H and orthogonal projections P, Q on H , (cid:107) f ( P, Q ) (cid:107) op = max σ ( P QP ) ψ f .
2. Define Ψ f : [0 , ∪ { , } → R by Ψ f | [0 , = ψ f and Ψ f (( l, k )) = | α lk | for ( l, k ) ∈ { , } . Denote σ P,Q = σ ( I − H ) ∪ Λ . Then for every separablecomplex Hilbert space H and orthogonal projections P, Q on H , (cid:107) f ( P, Q ) (cid:107) op = max σ P,Q Ψ f . Note that max [0 , ψ f = max [0 , ∪{ , } Ψ f , since ψ f (1) = max {| α | , | α |} and ψ f (0) = max {| α | , | α |} . The latter, together with Claim 2.15, immediately leads to the following.
Conclusion 2.9.
The constant M f = max [0 , ψ f = max [0 , ∪{ , } Ψ f is a universal, tight upper bound for (cid:107) f ( P, Q ) (cid:107) op , where P, Q are any orthogonalprojections on an arbitrary complex Hilbert space H . We conclude with the following example.
Example 2.10.
Let f ∈ ker( T ) where T is as in Remark 2.5, and recall that r ( t ) = (cid:112) t (1 − t ) . Then f ( P, Q ) = f ( P Q ) P Q − f ( QP ) QP for some univariate polynomial f , therefore g = g = 0 , g = − g ( t ) = f ( t ) r ( t ) . It follows that g ( t ) = 2 | g ( t ) | , g ( t ) = g ( t ) , hence we find that ψ f ( t ) = (cid:115) g ( t ) + (cid:112) g ( t ) − | g ( t ) | (cid:115) | g ( t ) | + (cid:112) | g ( t ) | − | g ( t ) | | g ( t ) | . Thus, (cid:107) f ( P, Q ) (cid:107) op = max σ ( I − H ) ( | f | r ) ≤ max [0 , ( | f | r ) = M f . . .3 The canonical form and angles between subspaces Assume that dim H = n . Recall the notation V = V P ∩ V Q , V = V P ∩ V ⊥ Q ,V = V ⊥ P ∩ V Q , V = V ⊥ P ∩ V ⊥ Q V = ( V ⊕ V ) ⊥ ∩ V P , V = ( V ⊕ V ) ⊥ ∩ V ⊥ P , and let m lk = dim V lk , l, k ∈ { , } and m = dim V = dim V . Definition 2.11.
Denote the eigenvalues of H by < µ ≤ ... ≤ µ m < . Thereduced principal angles < θ ≤ ... ≤ θ m < π associated with the pair ( P, Q ) are defined by sin θ l = µ l , l = 1 , ..., m. The pair (
P, Q ) is determined, up to unitary equivalence, by the numbers m , m , m , m , m together with the reduced principal angles. Definition 2.12.
Let m P = dim V P ≤ dim V Q = m Q (so m P = m + m + m ).The principal angles ≤ φ ≤ φ ≤ ... ≤ φ m P ≤ π of the pair ( P, Q ) are definedrecursively. The angle φ is specified by cos φ = max {|(cid:104) x, y (cid:105)| | x ∈ V P , y ∈ V Q , (cid:107) x (cid:107) = (cid:107) y (cid:107) = 1 } , and if cos φ = |(cid:104) x , y (cid:105)| , then cos φ = max {|(cid:104) x, y (cid:105)| | x ∈ V P ∩ { x } ⊥ , y ∈ V Q ∩ { y } ⊥ , (cid:107) x (cid:107) = (cid:107) y (cid:107) = 1 } . Next, if we denote V P,k = Span { x , ..., x k } and V Q,k = Span { y , ..., y k } , where cos ϕ l = |(cid:104) x l , y l (cid:105)| for l = 1 , ..., k , then cos φ k +1 = max {|(cid:104) x, y (cid:105)| | x ∈ V P ∩ V ⊥ P,k , y ∈ V Q ∩ V ⊥ Q,k , (cid:107) x (cid:107) = (cid:107) y (cid:107) = 1 } . The reduced principal angles are the principal angles lying in (cid:0) , π (cid:1) , i.e., φ = ... = φ m = 0 ,φ m +1 = θ , ..., φ m + m = θ m ,φ m + m +1 = ... = φ m + m + m = π . The following elementary examples will be of immediate use.
Example 2.13.
Let n ≥ and φ ∈ (cid:2) , π (cid:3) . For < m P ≤ m Q ≤ n − m P integers, there exists a pair of orthogonal projections P : C n → V P , Q : C n → V Q with m P = dim V P , m Q = dim V Q and smallest principal angle φ = φ .Indeed, if { e l | l = 1 , ..., n } is an orthonormal basis of C n , and V P = Span { e , ..., e m P } , then V Q = Span (cid:8) cos φe + sin φe n , e m P +1 , ..., e m P + m Q − (cid:9) satisfies the required. In particular, if φ ∈ (cid:0) , π (cid:1) , then sin φ ∈ σ ( H ) . xample 2.14. Let n ≥ , φ ∈ (cid:2) , π (cid:3) and assume that < m P ≤ m Q < n .Then there exists a pair of orthogonal projections P : C n → V P , Q : C n → V Q with m P = dim V P and m Q = dim V Q such that φ ∈ { φ , ..., φ m P } .Indeed, if m P + m Q ≤ n , we saw that it is possible to define P, Q such that φ = φ . If m P + m Q > n , then m = dim V P ∩ V Q > , so φ = ... = φ m = 0 .Thus if φ = 0 , we are done. Otherwise, we may set m = m P + m Q − n (notethat m < m P ), V P = Span { e , ..., e m P } and V Q = Span { e , ..., e m , cos φe m P + sin φe n , e m P +1 , ..., e n − } . Then
P, Q are as required.
The previous examples essentially amount to the proof of the following,where M f is the universal bound of Conclusion 2.9. Claim 2.15.
Let H be a separable complex Hilbert space with ≤ dim H ≤ ∞ .Let , Id (cid:54) = P : H → V P denote an orthogonal projection. Then for every ≤ m Q ≤ dim H − there exists an orthogonal projection Q : H → V Q with rank Q = m Q such that (cid:107) f ( P, Q ) (cid:107) op = M f .Proof. We need to find an orthogonal projection Q : H → V Q such that (cid:107) f ( P, Q ) (cid:107) op = M f . We assume without loss of generality that dim H = 2, so that dim V P = 1.Indeed, if we choose arbitrary non-zero v ∈ V P and v ∈ ker( P ), and denote H = Span { v , v } , and find an orthogonal projection Q : H → H such that (cid:107) f ( P | H , Q ) (cid:107) op = max [0 , ψ f , then clearly we can extend Q to H ⊥ so thatrank Q = m Q , which yields the required. Hence we assume dim H = 2.Let t max ∈ [0 ,
1] be such that max [0 , ψ f ( t ) = ψ f ( t max ). If 0 < t max < φ ∈ (cid:0) , π (cid:1) such that sin φ = 1 − t max . Clearly, the previousexamples imply that there exists Q : H → V Q such that φ is among the (reduced)principal angles associated with P, Q , which means that t max ∈ σ ( I − H ). Hence, (cid:107) f ( P, Q ) (cid:107) op = max [0 , ψ f .Otherwise, t max ∈ { , } , which means thatmax [0 , ψ f = max [0 , ∪{ , } Ψ f = max {| α | , | α | , | α | , | α |} . If the latter equals | α | or | α | , we may set Q = P , so that V P ∩ V Q = V P and V ⊥ P ∩ V ⊥ Q = V ⊥ P . Then (cid:107) f ( P, Q ) (cid:107) op = max {| α | , | α |} = max [0 , ψ f .Otherwise, we can set Q = I − P , so that V P ∩ V ⊥ Q = V P , V ⊥ P ∩ V Q = V ⊥ P toobtain the required. The present section is dedicated to the proof of Theorem 3.7, which includesTheorem 1.2 as a special case. The proof consists of little more than a straight-forward combination of results from [2] and [5]. Indeed, if f ∈ ker( T ), then by14onclusion 2.8, (cid:107) f ( P, Q ) (cid:107) op = max σ ( P QP ) ψ f (the general case f ∈ A is onlyslightly more complicated). The operator P QP ∈ End( V P ) is known to providea model of the so called 2-Jacobi ensemble of random matrices. In particular,the behavior of σ ( P QP ) is well understood as n → ∞ .Let Ω n,α,β = G a n ( n ) × G b n ( n ), equipped with the probability measure ν n,α,β = µ a n ,n × µ b n ,n . Here, a n = (cid:98) αn (cid:99) and b n = (cid:98) βn (cid:99) , where 0 < α ≤ β satisfy α + β ≤
1. For (
P, Q ) ∈ Ω n,α,β , we will use the notations of (4). Finally,let I fn,α,β = (cid:90) Ω n,α,β (cid:107) f ( P, Q ) (cid:107) op dν n,α,β . Our goal is to compute I fα,β = lim n →∞ I fn,α,β in terms of α, β and f , whereaccording to Fubini’s theorem, I fn,α,β = (cid:90) G an ( n ) (cid:32)(cid:90) G bn ( n ) (cid:107) f ( P, Q ) (cid:107) op dµ b n ,n ( Q ) (cid:33) dµ a n ,n ( P ) . (7)In this context, we note the following observation. Remark 3.1. (cid:107) f ( P, Q ) (cid:107) op ≤ M f for all ( P, Q ) ∈ Ω n,α,β by Conclusion 2.9,hence I fn,α,β ≤ M f . Also note that max Q ∈ G bn ( n ) (cid:107) f ( P, Q ) (cid:107) op = M f for every P ∈ G a n ( n ) , by Claim 2.15. The invariance of µ b n ,n implies that the inner integral in (7) is independentof the choice of P . To see this, let E n = { e , ..., e n } denote the standard basisof C n , and let P denote the orthogonal projection on Span { e , ..., e a n } . Thenfor every P ∈ G a n ( n ) there exists a (non-unique) unitary operator U such that P = U ∗ P U . Lemma 3.2.
Let Q ∈ G b n ( n ) . Then (cid:107) f ( P, Q ) (cid:107) op = (cid:107) f ( P , U QU ∗ ) (cid:107) op .Proof. It will be convenient to introduce the evaluation homomorphismsev(
P, Q ) :
A →
End( C n ) , specified by ev( P, Q )( f ) = f ( P, Q ). Thenev(
P, Q )( · ) = U ∗ ev( P , U QU ∗ )( · ) U . Indeed, ev(
P, Q )(1) = I = U ∗ ev( P , U QU ∗ )(1) U , ev( P, Q )( x ) = P = U ∗ P U = U ∗ ev( P , U QU ∗ )( x ) U and ev( P, Q )( y ) = Q = U ∗ ( U QU ∗ ) U = U ∗ ev ( P , U QU ∗ ) ( y ) U . This holds, similarly, for all the monomials in A , and by linearity, extends toall of A . Since U is unitary, we obtain the required.15he invariance of µ b n ,n implies that (cid:90) G bn ( n ) (cid:107) f ( P , U QU ∗ ) (cid:107) op dµ b n ,n ( Q ) = (cid:90) G bn ( n ) (cid:107) f ( P , Q ) (cid:107) op dµ b n ,n ( Q )for every unitary operator U on C n , hence we can conclude that I fn,α,β = (cid:90) G bn ( n ) (cid:107) f ( P , Q ) (cid:107) op dµ b n ,n ( Q ) . (8)Let H be the operator associated with the pair ( P , Q ) as in Theorem 2.2.According to Theorem 2.7, (cid:107) f ( P , Q ) (cid:107) op can be expressed conveniently using σ ( I − H ) ⊂ σ ( P QP ) (assuming that P , Q do not commute). Thus, we areled to consider the joint distribution of the eigenvalues of P QP . Theorem 3.3 ([7]) . The joint eigenvalue distribution of P QP ∈ End( V P ) in [0 , a n is given by d J n ( λ , ..., λ a n ) = 1 C n,α,β a n (cid:89) l =1 λ b n − a n l (1 − λ l ) n − ( a n + b n ) (cid:89) ≤ l 0. The limiting behavior of d J n that is relevant in our context is as follows. Theorem 3.4 ([7]) . If d J n is the joint distribution of ( λ , ..., λ a n ) ∈ [0 , a n ,then for F ∈ C [0 , , a n a n (cid:88) l = l F ( λ l ) P −→ (cid:90) F ( t ) d J ∞ ( t ) as n → ∞ , where d J ∞ ( t ) = 12 πα t (1 − t ) (cid:112) − ( t − λ − )( t − λ + ) [ λ − ,λ + ] dt, and λ ± = (cid:16)(cid:112) β (1 − α ) ± (cid:112) α (1 − β ) (cid:17) . Note that λ − < λ + , and λ − = 0 if andonly if α = β , and λ + = 1 if and only if α + β = 1 . Any n works if α = β , otherwise any n for which ( β − α ) n ≥ < λ − or λ + < 1, that asubset of order o ( a n ) of eigenvalues of P QP remains outside of [ λ − , λ + ], anda-priori it could be that µ b n ,n ( { Q | σ ( P QP ) ∩ ([0 , \ ( λ − − δ, λ + + δ )) (cid:54) = ∅} ) > ε for some ε, δ > 0. Hence, we will use the following. Theorem 3.5 ([5]) . For any compact set K such that K ∩ [ λ − , λ + ] = ∅ , thereexists C > such that µ b n ,n ( { Q | σ ( P QP ) ∩ K (cid:54) = ∅} ) < e − Cn . We conclude as follows. Conclusion 3.6. Let δ > . Denote A n,t,δ = { Q ∈ G b n ( n ) | d ( σ ( I − H ) , t ) < δ } ,B n,δ = { Q ∈ G b n ( n ) | σ ( I − H ) ⊂ ( λ − − δ, λ + + δ ) } . Then lim n →∞ µ b n ,n ( B n,δ ) = lim n →∞ µ b n ,n ( A n,t,δ ) = 1 .Proof. As we noted previously (9), we can assume that µ b n ,n ( G n, ) = 1. Thus,almost surely σ ( P QP ) = σ ( I − H ), so it is a basic consequence of the fact thatthe density of d J ∞ is non-vanishing on [ λ − , λ + ] that µ b n ,n ( A n,t,δ ) converges to1 as n → ∞ . Next, by Theorem 3.5,lim n →∞ µ b n ,n ( { Q ∈ G b n ( n ) | σ ( P QP ) ∩ K } ) = 0 , where K = [ − δ, λ − − δ ] ∪ [ λ + + δ, δ ] , hence lim n →∞ µ b n ,n ( B n,δ ) = 1.We derive Theorem 1.2 as a consequence of a more general result, as follows. Theorem 3.7. Let f ∈ A . Then I fα,β = lim n →∞ I fn,α,β is well defined, and1. If α = β = then I f,α,β = max [0 , ψ f = M f ,2. If α < β = 1 − α , then I f,α,β = max (cid:8) | α | , max [ λ − , ψ f (cid:9) ,3. If α = β < − α , then I f,α,β = max (cid:8) | α | , max [0 ,λ + ] ψ f (cid:9) ,4. If α < β < − α , then I f,α,β = max (cid:8) | α | , | α | , max [ λ − ,λ + ] ψ f (cid:9) .We recall that λ − = 0 ⇔ α = β and λ + = 1 ⇔ β = 1 − α (and that λ − < λ + ). Here, d ( K, t ) denotes the distance between the set K ⊂ R and t ∈ R . roof. As before, we use the notation of (4), and assume that µ b n ,n ( G n, ) = 1.Fix ε > 0. Then I fn,α,β = (cid:90) G bn ( n ) (cid:107) f ( P , Q ) (cid:107) op dµ b n ,n ( Q ) = (cid:90) G bn ( n ) (cid:18) max σ P ,Q Ψ f (cid:19) dµ b n ,n ( Q ) , where Ψ f and σ P ,Q are as in Conclusion 2.8. If dim V = 0 and dim V = 0,then max σ P ,Q Ψ f = max σ ( I − H ) ψ f . The latter observation will guide our proof,i.e., we will first show that˜ I fα,β = lim n →∞ (cid:90) G bn ( n ) (cid:18) max σ ( I − H ) ψ f (cid:19) dµ b n ,n ( Q ) = max [ λ − ,λ + ] ψ f . (10)Let t max ∈ [ λ − , λ + ] be such that M = max [ λ − ,λ + ] ψ f = ψ f ( t max ) . By the continuity of ψ f , there exists δ > | t − t max | < δ then | ψ f ( t ) − M | < ε , (11)and additionally, max [ λ − − δ,λ + + δ ] ∩ [0 , ψ f − M < ε . (12)Note that the latter is trivial if λ − = 0, λ + = 1 (since then the left hand sideequals 0). We will use (11) to show ˜ I fα,β ≥ M , and (12) to show ˜ I fα,β ≤ M .Using Conclusion 3.6, there exists n ∈ N such that for all n > n , (cid:16) M − ε (cid:17) µ b n ,n ( G n, ∩ A n,t max ,δ ) > M − ε. (13)Thus, since max σ P ,Q Ψ f ≥ max σ ( I − H ) ψ f ≥ 0, we get that for all n > n , M f ≥ I fn,α,β = (cid:90) G bn ( n ) (cid:18) max σ P ,Q Ψ f (cid:19) dµ b n ,n ≥ (cid:90) G n, ∩ A n,t max ,δ (cid:18) max σ ( I − H ) ψ f (cid:19) dµ b n ,n ≥ M − ε. The last inequality holds first since max σ ( I − H ) ψ f > M − ε for every projection Q ∈ G n, ∩ A n,t max ,δ by (11), and then using (13).If α = β = , then λ − = 0, λ + = 1, so M = M f and lim n →∞ I fn,α,β = M f as required. Otherwise, by Conclusion 3.6, for ε > M f ε < ε ,there exists n ∈ N such that for all n > n , it holds that µ b n ,n ( C n,δ ) > − ε , (14)18here C n,δ = G n, ∩ A n,t max ,δ ∩ B n,δ ,B n,δ = { Q ∈ G b n ( n ) | σ ( I − H ) ⊂ ( λ − − δ, λ + + δ ) } . Thus, (cid:90) G bn ( n ) (cid:18) max σ ( I − H ) ψ f (cid:19) dµ b n ,n = (cid:90) C n,δ (cid:18) max σ ( I − H ) ψ f (cid:19) dµ b n ,n + (cid:90) G n, ∩ (G bn ( n ) \ C n,δ ) (cid:18) max σ ( I − H ) ψ f (cid:19) dµ b n ,n ≤ M + ε ε M f < M + ε. Here, the first integral is not greater than M + ε since max σ ( I − H ) ψ f ≤ M + ε for every Q ∈ C n,δ using (12), and the second integral is bounded from aboveby ε M f since max σ ( I − H ) ψ f ≤ M f = max [0 , ψ f and then using (14).The above completes the proof of (10), and the proof for the case α = β = .We turn to address the remaining cases, namely, when α + β = 1 , α < β , andwhen α + β < , α = β and finally when α + β < , α < β . Note that for n large enough, if α < β then dim V > Q ∈ G b n ( n ), and if α + β < V > Q ∈ G b n ( n ).Assume that α + β = 1 and that α < β . Let M − = max {| α | , M } . If M − = | α | , then for n large enough,max σ P ,Q Ψ f = | α | for every Q ∈ G b n ( n ), hence clearly I fn,α,β = | α | for every n large enough.Otherwise, if | α | < M , we assume without loss of generality that M −| α | > ε .Then by the above, M − ε ≤ max σ P ,Q Ψ f = max σ ( I − H ) ψ f ≤ M + ε Q ∈ C n,δ , hence by the above, lim n →∞ I fn,α,β = M .Similarly, if α = β < , then either M + = max {| α | , M } = | α | , in whichcase we note that for n large enough, max σ P ,Q Ψ f = | α | for every Q ∈ G b n ( n )so I fn,α,β = | α | for every n large enough, or M + = M , in which case as for theprevious case, lim n →∞ I fn,α,β = M .Finally, if α + β < α < β , then either M − , + = max {| α | , | α | , M } =max {| α | , | α |} , in which case we note that for n large enough, max σ P ,Q Ψ f = M − , + for all Q ∈ G b n ( n ), hence I fn,α,β = M − , + for all n sufficiently large, orotherwise M − , + = M , in which case we repeat the reasoning above to obtainthe required. 19e obtain Theorem 1.2 immediately. Note that if α = β ≤ then (seeTheorem 3.4 for the formulas of λ − , λ + ) we obtain λ − = 0 , λ + = 4 α (1 − α ) = λ α . (15) Conclusion 3.8. If f ∈ ker( T ) , then by Lemma 2.3, | α lk | = 0 for l, k ∈ { , } ,hence the limits in the distinct cases addressed in Theorem 3.7 admit the sameformula, namely, lim n →∞ I fn,α,β = max [ λ − ,λ + ] ψ f . If α = β , then [ λ − , λ + ] = [0 , λ α ] , giving Theorem 1.2. The present section contains the proof of Theorem 1.4. In the next subsection,we prove (for the sake of completeness) some basic lemmas about weak andstrong convergence in Hilbert spaces. In subsection 4.2, we specify the limits of”central” matrix coefficients of the projections, as computed in a previous work([18]) by the author. The contents of subsections 4.1, 4.2 suffice to reduce (insubsections 4.3, 4.4) the evaluation of the limit to the case of two projectionson L ( T ) (where T ⊂ C is the unit circle), namely, the Cauchy-Szeg¨o projectionon the Hardy space (denoted Π T ), and the operator of multiplication by theindicator function of E α = { < (cid:60) ζ < α } ⊂ T (denoted M Eα ). Finally, insubsection 4.5, we use a classical result on the spectrum of Toeplitz operatorswith bounded symbols (together with Theorem 2.7) to conclude the proof. Let B ( H ) denote the space of bounded operators on a complex Hilbert space H .We will use the following elementary notions and facts. Definition 4.1. Let { A d } d ∈ N ⊂ B ( H ) and A ∈ B ( H ) .1. We say that { A d } d ∈ N converges strongly to A if lim d →∞ A d v = Av forevery v ∈ H .2. We say that { A d } d ∈ N converges weakly to A if lim d →∞ (cid:104) A d u, v (cid:105) = (cid:104) Au, v (cid:105) for every u, v ∈ H . Lemma 4.2. Let E = { e k | k ∈ N } denote an orthonormal basis of H . Let { A d } d ∈ N ⊂ B ( H ) and A ∈ B ( H ) . Then { A d } d ∈ N converges weakly to A if andonly if sup d (cid:107) A d (cid:107) op < ∞ and lim d →∞ (cid:104) A d e l , e k (cid:105) = (cid:104) Ae l , e k (cid:105) for every l, k ∈ N .Proof. It is well known that weakly convergent sequences are bounded. Indeed,for v ∈ H let ψ d,v ( u ) = (cid:104) A d v, u (cid:105) , ψ d,v ∈ H ∗ . Then {| ψ d,v ( u ) |} d ∈ N is bounded forevery u ∈ H , hence {(cid:107) ψ d,v (cid:107) H ∗ } d ∈ N = {(cid:107) A d v (cid:107) H } d ∈ N is bounded by the uniformboundedness principle. Since {(cid:107) A d v (cid:107)} d ∈ N is bounded for all v ∈ H , then againusing uniform boundedness, we deduce that {(cid:107) A d (cid:107) op } d ∈ N is bounded.20onversely, assume that (cid:107) A d (cid:107) op < M for all d , let u, v ∈ H and let ε > δ > 0, we may take˜ u = (cid:88) k ≤ k (cid:104) u, e k (cid:105) e k , ˜ v = (cid:88) k ≤ k (cid:104) v, e k (cid:105) e k such that (cid:107) u − ˜ u (cid:107) < δ, (cid:107) v − ˜ v (cid:107) < δ . Then, for sufficiently large d so that |(cid:104) ( A d − A )˜ u, ˜ v (cid:105)| < δ , we have that |(cid:104) A d u, v (cid:105) − (cid:104) Au, v (cid:105)| = | ( A d − A )( u − ˜ u ) , v (cid:105) + (cid:104) ( A d − A )˜ u, v − ˜ v (cid:105) + (cid:104) ( A d − A )˜ u, ˜ v (cid:105)|≤ (cid:107) A d − A (cid:107) op ( (cid:107) v (cid:107) + (cid:107) ˜ u (cid:107) ) δ + δ ≤ ( M + (cid:107) A (cid:107) op )( (cid:107) v (cid:107) + (cid:107) u (cid:107) ) δ + δ < ε, where the latter holds for δ sufficiently small. Lemma 4.3. Let E = { e k | k ∈ N } denote an orthonormal basis of H . If { A d } d ∈ N ⊂ B ( H ) converges weakly to A ∈ B ( H ) and lim d →∞ (cid:107) A d e k (cid:107) = (cid:107) Ae k (cid:107) for every (fixed) k ∈ N , then A d converges strongly to A .Proof. We note that (cid:107) A d e k − Ae k (cid:107) = (cid:107) A d e k (cid:107) + (cid:107) Ae k (cid:107) − (cid:104) Ae k , A d e k (cid:105) − (cid:104) A d e k , Ae k (cid:105) , hence by weak convergence,lim d →∞ (cid:107) A d e k − Ae k (cid:107) = 0 , that is, lim d →∞ A d e k = Ae k for every fixed k ∈ N .Let v = (cid:80) k ∈ N v k e k ∈ H . Let ε > 0. Note that sup d (cid:107) A d (cid:107) op < ∞ . Hencethere exists k > v ε = (cid:80) k ≥ k v k e k satisfies (cid:107) ( A − A d ) v ε (cid:107) < ε . Let u ε = v − v ε . Then lim d →∞ A d u ε = Au ε , so there exists d such that for all d ≥ d , it holds that (cid:107) ( A d − A ) u ε (cid:107) < ε . Thusfor all d ≥ d , 0 ≤ (cid:107) ( A d − A ) v (cid:107) = (cid:107) ( A d − A )( u ε + v ε ) (cid:107) < ε, as required. Lemma 4.4. Assume that { A d } d ∈ N ⊂ B ( H ) converges to A ∈ B ( H ) strongly.Then lim inf (cid:107) A d (cid:107) op ≥ (cid:107) A (cid:107) op (this is also true if A d converges to A weakly).Proof. Let ε > 0. Assume that v ∈ H satisfies (cid:107) Av (cid:107) > (cid:107) A (cid:107) op − ε . Then thereexists d such that for all d > d , it holds that (cid:107) A d v (cid:107) > (cid:107) A (cid:107) op − ε . Hence therequired. Conclusion 4.5. If { A d } d ∈ N ⊂ B ( H ) converges to A ∈ B ( H ) strongly and (cid:107) A d (cid:107) op ≤ (cid:107) A (cid:107) op for all d then lim d →∞ (cid:107) A d (cid:107) op = (cid:107) A (cid:107) op . emma 4.6. Assume that { A d } d ∈ N and { B d } d ∈ N converge to A, B strongly.Then A d B d converges strongly to AB .Proof. Let M A > (cid:107) A d (cid:107) op < M A for all d . Let v ∈ H . Then (cid:107) ( A d B d − AB ) v (cid:107) = (cid:107) ( A d B d − A d B + A d B − AB ) v (cid:107)≤ (cid:107) A d ( B d − B ) v (cid:107) + (cid:107) ( A d − A ) Bv (cid:107)≤ M A (cid:107) ( B d − B ) v (cid:107) + (cid:107) ( A d − A ) Bv (cid:107) , hence by strong convergence of A d , B d , lim d →∞ (cid:107) ( A d B d − AB ) v (cid:107) = 0. We consider the standard basis B ,d = { e (3) j , e (3) j − , ..., e (3) − j } of eigenvectors of J ,where as before 2 j + 1 = d . Recall that the spectrum of J , J , J is σ d = { j, j − , ..., − j } , and denote P ,α,d = (0 ,α d ) ( J ) and P ,α,d = (0 ,α d ) ( J ), where α d > , α d ) ∩ σ d ) = (cid:98) αd (cid:99) for every d ∈ N . Let [ A ] B ,d denote the matrix of A : C d → C d relative to B ,d .Clearly, [ P ,α,d ] B ,d = (cid:18) ˜ I α,d 00 0 (cid:19) , (16)where ˜ I α,d = (cid:18) I (cid:98) αd (cid:99) (cid:19) . Write [ P ,α,d ] B ,d = ( P ,α,d,m (cid:48) ,m ) m (cid:48) ,m = j,j − ,..., − j . Theorem 4.7 ([18]) . Fix m (cid:48) , m ∈ σ d . Let ˆ E α ( k ) denote the k -th Fouriercoefficient of the indicator function of E α = { < (cid:60) ζ < α } ⊂ T . Then lim k →∞ P ,α,d +2 k,m (cid:48) ,m = ˆ E α ( m − m (cid:48) ) . In particular, Conclusion 4.8. Fix m ∈ σ d . Then lim k →∞ (cid:13)(cid:13)(cid:13) P ,α,d +2 k e (3) m (cid:13)(cid:13)(cid:13) = ˆ E α (0) .Proof. Note that (cid:104) P ,α,d e (3) m , P ,α,d e (3) m (cid:105) = (cid:104) P ∗ ,α,d P ,α,d e (3) m , e (3) m (cid:105) = (cid:104) P ,α,d e (3) m , e (3) m (cid:105) = P ,α,d,m,m . Thus, by Theorem 4.7, we obtain the required.22 .3 The l ( Z ) settings It will be convenient to work in L ( T ) rather than C d . As an intermediate step,we consider l ( Z ). Let ˆ B = { ˆ e k | k ∈ Z } denote the standard basis of l ( Z ).Define an embedding Ψ d : C d → l ( Z ) byΨ d (cid:16) e (3) m (cid:17) = (cid:26) ˆ e m − d ∈ N ˆ e m − d ∈ N − , m = j, j − , ..., − j, d = 2 j + 1 . Let V d = Ψ d (cid:0) C d (cid:1) , and let Π d : l ( Z ) → V d ⊂ l ( Z ) denote the orthogonalprojection on V d . For an operator A d : C d → C d we set A Z d = Ψ d ◦ A d ◦ Ψ − d ◦ Π d . Conclusion 4.9. If the matrix of A d in B ,d is ( a m (cid:48) ,m ) | m | , | m (cid:48) |≤ j , then thematrix elements of A Z d in the basis ˆ B are as follows.If d ∈ N − , then (cid:104) A Z d ˆ e l , ˆ e k (cid:105) = (cid:26) | l + 1 | > j or | k + 1 | > j,a k +1 ,l +1 | l + 1 | , | k + 1 | ≤ j If d ∈ N , then (cid:104) A Z d ˆ e l , ˆ e k (cid:105) = (cid:26) (cid:12)(cid:12) l + (cid:12)(cid:12) > j or (cid:12)(cid:12) k + (cid:12)(cid:12) > ja k + ,l + (cid:12)(cid:12) l + (cid:12)(cid:12) , (cid:12)(cid:12) k + (cid:12)(cid:12) ≤ j L ( T ) settings Let B = { ζ k | k ∈ Z } denote the standard orthonormal basis of L ( T ), whichwe identify with l ( Z ) in the obvious way. Let P T ,α,d , P T ,α,d be the equivalentsof P Z ,α,d , P Z ,α,d . Let Π T : L ( T ) → H ( T ) be the orthogonal Cauchy-Szeg¨oprojection on the Hardy space H ( T ) ⊂ L ( T ). Finally, for ψ ∈ L ∞ ( T ) let M ψ : L ( T ) → L ( T ) be the multiplication operator G (cid:55)→ ψG . Claim 4.10. P T ,α,d , P T ,α,d converge strongly to Π T , M Eα respectively.Proof. Clearly (cid:107) P T ,α,d (cid:107) op = (cid:107) P T ,α,d (cid:107) op = 1 for all d ∈ N . By (16), Claim 4.7and Conclusion 4.9, we have for every k, l ∈ Z lim d →∞ (cid:104) P T ,α,d ζ l , ζ k (cid:105) L ( T ) = (cid:104) Π T ζ l , ζ k (cid:105) L ( T ) , lim d →∞ (cid:104) P T ,α,d ζ l , ζ k (cid:105) L ( T ) = (cid:104)M Eα , ζ l , ζ k (cid:105) L ( T ) . Thus, Lemma 4.2 gives us weak convergence. Then, as evident in (16),lim d →∞ P T ,α,d ζ k = Π T ζ k for all k ∈ Z . Also, (cid:13)(cid:13) M Eα ζ k (cid:13)(cid:13) = (cid:104) E α ζ k , E α ζ k (cid:105) L ( T ) = (cid:104) E α , (cid:105) L ( T ) = ˆ E α (0) . { P T ,α,d } d ∈ N and { P T ,α,d } d ∈ N . Conclusion 4.11. f ( P T ,α,d , P T ,α,d ) converges strongly to f (Π T , M Eα ) .Proof. This follows for all monomials by induction using Lemma 4.6, then forall polynomials using linearity. Π T , M Eα We study P = Π T , Q = M Eα in the context of the general theory of pairs oforthogonal projections. In the notations of Theorem 2.2, P QP = (1 , , , ⊕ (cid:18) I − H 00 0 (cid:19) . Definition 4.12. The Toeplitz operator associated with the symbol φ ∈ L ∞ ( T ) is T φ = Π T M φ Π T : H ( T ) → H ( T ) . Thus, P QP = (1 , , , ⊕ ( I − H ) ⊕ T Eα . We note the following classical result. Theorem 4.13 (Hartman-Wintner, [11], [6], 7.20) . If φ ∈ L ∞ ( T ) is real-valued,then the spectrum of T φ is given by σ ( T φ ) = [ess inf φ, ess sup φ ] . Conclusion 4.14. Hartman-Wintner’s Theorem implies that σ ( I − H ) = σ ( H ) = [0 , . Thus, by Theorem 2.7 and Conclusion 2.9, (cid:107) f ( P, Q ) (cid:107) op = M f . The polynomial f ( P T ,α,d , P T ,α,d ) converges to f ( P, Q ) strongly, hence by Conclusion 4.5 lim d →∞ (cid:107) f ( P T ,α,d , P T ,α,d ) (cid:107) op = M f , and since (cid:107) f ( P ,α,d , P ,α,d ) (cid:107) op = (cid:107) f ( P T ,α,d , P T ,α,d ) (cid:107) op , this completes the proofof Theorem 1.4. Remark 4.15. By the F. and M. Riesz Theorem, if G is holomorphic on theunit disk D ⊂ C , with zero radial boundary values on a subset E ⊂ T of positiveLebesgue measure, then G ≡ . Thus, the same holds for an anti-holomorphicfunction. Hence (in the notation of (4)), for the projections P, Q , we have V lk = { } for every l, k ∈ { , } , so that in fact P QP = T Eα = I − H : H ( T ) → H ( T ) . Concluding remarks The proof of Theorem 1.4 relies on the fact that in the semiclassical limit d → ∞ ,the spectral projections P ,α,d and P ,α,d converge to M Eα and Π T in someappropriate sense. Analogous facts hold for the rest of the pairs of spectralprojections studied in [18], hence the proof can be adapted so as to apply tothem as well.Notably, we considered pairs of spectral projections coming from positionand momentum operators ˆ q = M q , ˆ p = − i (cid:126) ∂∂q on L ( R ). We also considered pairs of spectral projections corresponding to theoperators cos ˆ θ = M cos θ and cos ˆ L , whereˆ θ = M θ , ˆ L = − i πn ∂∂θ are the analogues ([13, 15]) of ˆ q , ˆ p on L ( T ) (cid:39) L (cid:0) [0 , π ] , dθ π (cid:1) . Yet anotherexample involved the generators g , g of the finite Heisenberg groups H ( Z n ) ([17, 21, 20]), which act on F ∈ l ( Z n ) ≈ L ( T ) by g F ( k ) = e πkn i F ( k ) , g F ( k ) = F ( k + 1) . Let E = { ζ ∈ T | (cid:60) ζ > } . Then the pair of projections (0 , ∞ ) (ˆ q ) = M (0 , ∞ ) , (0 , ∞ ) (ˆ p )is unitarily equivalent to M E , Π T independently of (cid:126) (since (0 , ∞ ) (ˆ p ) is justthe projection on the Hardy space H ( R )). For the pair (0 , ∞ ) (cos ˆ θ ) = M E , (0 , ∞ ) (cos ˆ L ) , a sequence of unitary operators U n : L ( T ) → L ( T ) is required in order toreduce the proof to the case of M E and Π T . Similarly, forΠ = (0 , ∞ ) ( (cid:60) g ) , Π = (0 , ∞ ) ( (cid:60) g ) , (17)where (cid:60) A = ( A + A ∗ ), we need a sequence of embeddings ˜ U n : l ( Z n ) → L ( T ). U n , ˜ U n are defined by mapping elements of the standard bases of L ( T ), l ( Z n )to those of L ( T ) in a certain suitable way, so as to obtain the convergenceof the relevant spectral projections to the pair Π T , M E . The arguments areessentially the same as those of subsections 4.3, 4.4. Finally, applying Conclusion4.14 to Π T , M E , we obtain the following. Here, Z n = Z /n Z . onclusion 5.1. Analogues of Theorem 1.4 hold for the families of pairs ofspectral projections detailed above, i.e., lim n →∞ (cid:107) f (Π , Π ) (cid:107) op = lim n →∞ (cid:13)(cid:13)(cid:13) f ( M E , (0 , ∞ ) (cos ˆ L )) (cid:13)(cid:13)(cid:13) op = (cid:13)(cid:13) f ( M (0 , ∞ ) , (0 , ∞ ) (ˆ p ) (cid:13)(cid:13) op = M f for every f ∈ A . Let H be a complex Hilbert space with dim H < ∞ . Assume that P : H → V P and Q : H → V Q are two orthogonal projections with dim V P ≤ dim V Q . Letus for now consider P QP as an element of End( V P ). Then the canonical form(Theorem 2.2) implies that P QP = Id V ⊕ V ⊕ (Id − H ) . Thus (see subsection 2.3), the eigenvalues λ ≥ ... ≥ λ dim V P of P QP are givenby λ k = cos φ k , where φ ≤ ... ≤ φ dim V P are the principal angles between thesubspaces V P , V Q . Conclusion 5.2. If f ∈ ker( T ) , then using Conclusion 2.8, (cid:107) f ( P, Q ) (cid:107) op = max σ ( P QP ) ψ f = max φ ∈ Φ ψ f (cos φ ) , where Φ is the set of principal angles between V P , V Q . The next images illustrate the striking discrepancy between the spectrumof random P QP ∈ End( V P ), and that of P ,α,d P ,α,d P ,α,d ∈ End (Im ( P ,α,d )),even when α = , i.e., when the limits (2), (3) coincide for every f ∈ A .Figure 7: The sorted eigenvalues of P QP ∈ End ( V P ), where P, Q ∈ G n (2 n )are random and n = 1000. 26igure 8: The sorted eigenvalues of P ,α,d P ,α,d P ,α,d ∈ End (Im ( P ,α,d )) for d = 2000 and α = . The number of values that are visibly between 0 and 1seems to grow very slowly as d → ∞ .The eigenvalues of P ,α,d P ,α,d P ,α,d appear to cluster near 0, 1 for every0 < α ≤ , but this phenomenon is unproven. By contrast, the very samephenomenon is rather well known in the case of the projections Π , Π specifiedin (17). Namely, Theorem 5.3 ([8]) . Let < ε < . Let S n ( a, b ) be the number of eigenvaluesof Π Π Π ∈ End ( Im (Π )) in the interval ( a, b ) . Then S n (0 , ε ) , S n (1 − ε, ∼ n , S n ( ε, − ε ) = O (log n ) . This is essentially a special case of a result from [8]. The latter also seems to hold for spectral projections of spin operators. Conjecture. Let < ε < . Let R d ( a, b ) be the number of eigenvalues of P ,α,d P ,α,d P ,α,d ∈ End ( Im ( P ,α,d )) in the interval ( a, b ) . Then as d → ∞ , R d (0 , ε ) ∼ c α αd, R d (1 − ε, ∼ (1 − c α ) αd, R d ( ε, − ε ) = O (log d ) , for some constant < c α < (with c = ). The above suggests that the rate of convergence in Theorem 1.4 (and inConclusion 5.1, for Π , Π ) is much slower than in Theorem 1.2. Indeed, considerthe case of commutators ( f ( x, y ) = xy − yx ), depicted in figures 1, 2. Then ψ f ( t ) = (cid:112) t (1 − t ) by Example 1.7 and ψ f (cos φ ) = sin φ cos φ . Thus, byConclusion 5.2, (cid:107) [ P, Q ] (cid:107) op = max φ ∈ Φ sin φ cos φ = sin φ cos φ , where φ ∈ Φ is the principal angle closest to π .27 onclusion 5.4. Let Φ d denote the set of principal angles between Im ( P ,d ) and Im ( P ,d ) . While Theorem 1.4 implies that Φ d becomes dense in (cid:2) , π (cid:3) as d → ∞ , Figure 8 suggests that this happens very slowly (at a logarithmic pace,conjecturally). Thus, min φ ∈ Φ d | φ − π | tends to slowly, hence (cid:107) [ P ,d , P ,d ] (cid:107) op converges to slowly.By contrast, Figure 7 suggests that the set of principal angles Φ associatedwith a random pair P, Q ∈ G (cid:98) n (cid:99) ( n ) is distributed much more evenly, therefore min φ ∈ Φ | φ − φ | tends to quite quickly for every φ ∈ (cid:2) , π (cid:3) and in particularfor φ = π , hence typically (cid:107) [ P, Q ] (cid:107) op converges quickly to .Similarly for P, Q ∈ G (cid:98) αn (cid:99) ( n ) , P ,α,d , P ,α,d and general f ∈ ker( T ) . Let us offer an informal, conjectured explanation for ”maximality results” ofthe type of Theorem 1.4 and Conclusion 5.1. The explanation is based on thenotion of quantization, and is inspired by findings from [14, 4]. In what follows, L ( H ) denotes the space of self-adjoint operators on a Hilbert space H .Let ( M, ω ) denote a closed , quantizable symplectic manifold. A Berezin-Toeplitz quantization ([3, 12, 16]) of M produces a sequence of finite dimensionalcomplex Hilbert spaces H d , such that lim d →∞ dim H d = + ∞ , together withsurjective linear maps T d : C ∞ ( M ) → L ( H d ). The maps are required to satisfyseveral desirable properties in the semiclassical limit d → ∞ . Example 5.5. Up to normalization, J = T d ( x ) and J = T d ( x ) , where x , x : S → R are the Cartesian coordinate functions. Let F , F ∈ C ∞ ( M ) and assume that I , I ⊂ R are a pair of non-trivialintervals. We consider the spectral projectionsΠ ,d = I ( T d ( F )) , Π ,d = I ( T d ( F ))as a pair of quantum observables that are somehow related ([22]) to the domains D = F − ( I ) , D = F − ( I ) . Example 5.6. In this interpretation, P ,α,d , P ,α,d are ”associated” with thedomains { < x ≤ α } , { < x ≤ α } ⊂ S . Recall that M f denotes the universal, tight upper bound for (cid:107) f ( P, Q ) (cid:107) op ,where P, Q are arbitrary orthogonal projections on a separable complex Hilbertspace. Our various numerical simulations appear to support the following. Conjecture. Fix f ∈ ker( T ) . Assume that F , F are Poisson non-commuting.Then1. If ∂D ∩ ∂D (cid:54) = ∅ is transversal, then lim d →∞ (cid:107) f (Π ,d , Π ,d ) (cid:107) op = M f . i.e., compact and without boundary. i.e., ω π represents an integral de-Rham cohomology class. . If the distance between ∂D , ∂D is greater than some ε > , then lim d →∞ (cid:107) f (Π ,d , Π ,d ) (cid:107) op = 0 . Note that this is essentially a conjecture about the principal angles betweensubspaces spanned by eigenstates of quantum observables (see Conclusion 5.2).We refer the reader to [18] (the final section in particular) for further detailsand simulations (mostly involving spin operators, but also some simulations forfinite Heisenberg groups). Acknowledgements This research has been partially supported by the European ResearchCouncil Starting Grant 757585 and by the Israel Science Foundation grant1102/20. I wish to express my sincere gratitude to the European ResearchCouncil and to the Israel Science Foundation.I wish to thank my advisors Leonid Polterovich and Lev Buhovski formany meetings and discussions, useful comments, and general guidancein this project. I wish to thank Boaz Klartag, Mikhail Sodin and SashaSodin for useful discussions and comments.Finally, I wish to thank David Kazhdan for his suggestion to considerthe topics addressed in this work, and for providing the conjecture whichinitiated this project. The findings presented here would not have beenpossible without his involvement and insight. References [1] L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Physics:Theory and Application , Encyclopedia of Mathematics and its Applications, , Addison-Wesley Publishing Company, Reading, MA, (1981).[2] A. B¨ottcher, I.M. Spitkovsky, A gentle guide to the basics of two projectionstheory , Linear Algebra Appl. (2010), 1412-1459.[3] L. Charles, Quantization of Compact Symplectic Manifolds , J. Geom. Anal. (2016), 2664-2710.[4] L. Charles, L. Polterovich, Sharp correspondence principle and quantummeasurements . Algebra i Analiz (2017), no. 1, 237-278[5] B. Collins, Product of random projections, Jacobi ensembles and universalityproblems arising from free probability , Probab. Theory Related Fields (2005), 315-344.[6] R. G. Douglas, Banach Algebra Techniques in Operator Theory , GraduateTexts in Mathematics, Springer-Verlag, NY, 2nd edition, 1998[7] I. Dumitriu, E. Paquette, Global fluctuations for linear statistics of β -Jacobiensembles , Random Matrices Theory Appl. (2012), 1250013298] A. Edelman, P. McCorquodale, S. Toledo, The future fast Fourier transform? SIAM J. Sci. Comput. , no. 3 (1998), 1094-1114[9] R. Giles, H. Kummer, A matrix representation of a pair of projections in aHilbert space , Canad. Math. Bull (1971), 35-44[10] P. Halmos, Two subspaces , Trans. Amer. Math. Soc. (1969) 381-389[11] P.Hartman, A. Wintner, The spectra of Toeplitz’s matrices , Amer. J. Math. (1954), 867-882[12] Y. Le Floch, A Brief Introduction to Berezin-Toeplitz Operators on Com-pact K¨ahler Manifolds , CRM Short Courses, Springer International Pub-lishing, 2014.[13] N. Mukunda, Wigner distribution for angle coordinates in quantum me-chanics . American Journal of Physics , 182 (1979).[14] L. Polterovich, Symplectic geometry of quantum noise . Commun. Math.Phys. (2014), 481-519.[15] M.A. Przanowski, J. Tosiek, Remarks on Deformation Quantization OnThe Cylinder . Acta Physica Polonica B (2000) 561-587.[16] M. Schlichenmaier, Berezin-Toeplitz quantization for compact K¨ahler man-ifolds. A review of results , Adv. Math. Phys. (2010), Article ID 927280,doi:10.1155/2010/927280.[17] J. Schwinger, Unitary operator bases . Proc. Nat. Acad. Sci. USA (1960),570-579.[18] O. Shabtai, Commutators of spectral projections of spin operators ,arXiv:2008.00221[19] I.M. Spitkovsky, Once more on algebras generated by two projections , Lin-ear Algebra Appl. (1994), 377-395[20] V.S. Varadarajan, D. Weisbard, Convergence of quantum systems on grids .J. Math. Anal. Appl. (2007), 608-624.[21] A. Vourdas, Quantum systems with finite Hilbert space . Rep. Prog. Phys. (2004), 267-320.[22] S. Zelditch, P. Zhou, Central Limit Theorem for Spectral Partial BergmanKernels , Geom. Topol., (2019), 1961–2004.