Multiplicities, invariant subspaces and an additive formula

doi:10.1017/S0013091521000146

MULTIPLICITIES, INVARIANT SUBSPACES AND AN ADDITIVE FORMULA*

ARUP CHATTOPADHYAY¹, JAYDEB SARKAR² AND SRIJAN SARKAR³

1Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India(arupchatt@iitg.ac.in,2003arupchattopadhyay@gmail.com)

2Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India(jay@isibang.ac.in,jaydeb@gmail.com)

3Department of Mathematics, Indian Institute of Science, Bangalore 560012, India (srijans@iisc.ac.in,srijansarkar@gmail.com)

(Received 02 June 2020; first published online 30 April 2021)

Abstract LetT= (T1, . . . , Tn) be a commuting tuple of bounded linear operators on a Hilbert space H. The multiplicity ofT is the cardinality of a minimal generating set with respect toT. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Letn≥2, and letQi,i= 1, . . . , n, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc inC. IfQ^⊥_i,i= 1, . . . , n, is a zero-based shift invariant subspace, then the multiplicity of the jointMz = (Mz1, . . . , Mzn)-invariant subspace (Q1⊗ · · · ⊗ Qn)^⊥ of the Dirichlet space or the Hardy space over the unit polydisc inCⁿ is given by

mult_Mz|

(Q1⊗···⊗Qn)⊥(Q1⊗ · · · ⊗ Qn)^⊥= n i=1

(mult_Mz|

Q⊥i

(Q^⊥_i)) =n.

A similar result holds for the Bergman space over the unit polydisc.

Keywords:Hardy space; Dirichlet space; Bergman and weighted Bergman spaces; polydisc; rank;

multiplicity; joint invariant subspaces; semi-invariant subspaces; zero-based invariant subspaces; tensor product Hilbert spaces

2020Mathematics subject classifications Primary 47A13; 47A15; 47A16; 47A80; 47B37; 47B38;

Secondary 46C99; 32A35; 32A36; 32A70

1. Introduction

This paper is concerned with an additive formula for a numerical invariant of commuting tuples of bounded linear operators on Hilbert spaces. The additive formula arises naturally in connection with a class of simple invariant subspaces of the two-variable Hardy space

*Dedicated to Professor Kalyan Bidhan Sinha on the occasion of his 75th birthday.

H²(D²) [5]. From the function Hilbert space point of view, our additive formula is more refined for zero-based invariant subspaces of the Dirichlet space, the Hardy space, the Bergman space and the weighted Bergman spaces over the open unit polydiscDⁿ inCⁿ. To be more specific, let us first define the numerical invariant. Given an n-tuple of commuting bounded linear operatorsT := (T₁, . . . , T_n) on a Hilbert spaceH, we denote by

mult_T(H) = min{#G: [G]_T =H, G⊆ H}, where

[G]_T = span{T^k(G) :k ∈Zⁿ₊}, andT^k =T₁^k1· · ·T_n^kn for allk= (k₁, . . . , k_n)∈Zⁿ₊. If

mult_T(H) =m <∞,

then we say that themultiplicity of T ism. One also says that T is m-cyclic. Ifm= 1, then we also say thatT iscyclic. A subsetG ofH is said to begenerating subset with respect to T if [G]_T =H.

We pause to note that the computation of multiplicities of (even concrete and simple) bounded linear operators is a challenging problem (perhaps due to its inherent dynamical nature). We refer Rudin [17] for concrete (as well as pathological) examples of invariant subspaces of H²(D²) of infinite multiplicities and [4, 5, 11–13] for some definite results on computations of multiplicities (also see [7]).

The following example, as hinted above, illustrates the complexity of computations of the multiplicities of general function Hilbert spaces. As a first step, we consider the Hardy spaceH²(D) overD(the space of all square summable analytic functions onD) and the multiplication operatorM_z by the coordinate functionz. LetSbe a closedM_z-invariant subspace of H²(D). Then Q=S^⊥ is a closedM_z^∗-invariant subspace of H²(D). It then follows from Beurling that

mult_Mz|S(S) = 1,

that is, M_z|S on S is cyclic. Moreover, taking into account that mult_Mz(H²(D)) = 1, we obtain (cf. Proposition2.1)

mult_PQMz|Q(Q) = 1,

whereP_Q denotes the orthogonal projection ofH²(D) onto Q.

Now we consider the commuting pair of multiplication operatorsM_z = (M_z1, M_z2) on H²(D²) (the Hardy space over the bidisc). Observe thatH²(D²)∼=H²(D)⊗H²(D). Let Q1andQ2be two non-trivial closedM_z^∗-invariant subspaces ofH²(D). ThenQ1⊗ Q2is a joint (M_z^∗₁, M_z^∗₂)-invariant subspace ofH²(D²), and so (Q1⊗ Q2)^⊥ is a joint (M_z1, M_z2)- invariant subspace of H²(D²). Set M_z|_(Q1⊗Q2)^⊥ = (M_z1|_(Q1⊗Q2)^⊥, M_z2|_(Q1⊗Q2)^⊥). An equivalent reformulation of Douglas and Yang’s question (see page 220 in [6] and also [5]) then takes the following form: Is

mult_Mz|

(Q1⊗Q2)⊥(Q₁⊗ Q₂)^⊥= 2?

The answer to this question is yes and was obtained by Das along with the first two authors in [5]. This result immediately motivates (see page 1186, [5]) the following natural

question: Consider the joint M_z = (M_z1, . . . , M_zn)-invariant subspace (Q1⊗ · · · ⊗ Qn)^⊥ of H²(Dⁿ) where Q1, . . . ,Qn are non-trivial closed M_z^∗-invariant subspaces of H²(D).

Is then

mult_Mz|

(Q1⊗···⊗Qn)⊥(Q₁⊗ · · · ⊗ Q_n)^⊥=n?

This can be reformulated more concretely as follows: Let H_i be the Dirichlet space, the Hardy space, the Bergman space, or the weighted Bergman spaces overD(or, more gener- ally, a reproducing kernel Hilbert spaces of analytic functions onDfor which the operator M_zof multiplication by the coordinate functionzonHiis bounded),i= 1, . . . , n. Suppose Q^⊥_i is anM_z-invariant closed subspace ofHi,i= 1, . . . , n. Is then

mult_Mz|(Q

1⊗···⊗Qn)⊥(Q1⊗ · · · ⊗ Qn)^⊥ = n

i=1

(mult_Mz|

Q⊥i (Q^⊥_i ))?

In this paper, we aim to propose an approach to verify the above equality for a large class of function Hilbert spaces over Dⁿ. The methods and techniques used in this paper are completely different from [5] and can also be applied for proving more powerful results in the setting of general Hilbert spaces. There is indeed a more substantial answer, valid in a larger context of tensor products of Hilbert spaces (see Theorem4.3).

LetH ⊆ O(D) be a reproducing kernel Hilbert space and let the operatorM_zis bounded on H. Suppose S is aM_z-invariant closed subspace ofH. We say that S is azero-based invariant subspace if there existsλ∈Dsuch thatf(λ) = 0 for allf ∈ S.

A particular case of our main theorem is the following: Let H_i be the Dirichlet space, the Hardy space, the Bergman space, or the weighted Bergman spaces over D. LetS_i be an M_z^∗-invariant closed subspace of H_i, i= 1, . . . , n. SupposeS_i :=Q^⊥_i is a zero-based M_z-invariant closed subspace ofH_i such that

dim(S_izS_i)<∞ and [S_izS_i]_Mz|S

i =S_i, for alli= 1, . . . , n, then

mult_Mz|

(Q1⊗···⊗Qn)⊥(Q₁⊗ · · · ⊗ Q_n)^⊥= n i=1

(mult_Mz|

Q⊥i (Q^⊥_i )) = n i=1

dim(S_izS_i).

Note that the finite dimensional and generating subspace assumptions are automatically satisfied ifHi is the Hardy space or the Dirichlet space. However, ifSis anM_z-invariant closed subspace of the Bergman space overD, then

dim(S zS)∈N∪ {∞}.

We refer the reader to [2,8, 9] for more information. See also [10] for related results in the setting of weighted Bergman spaces overD.

The proof of the above additivity formula uses generating wandering subspace property, geometry of (tensor product) Hilbert spaces, and subspace approximation technique.

The paper is organized as follows. In §2, we set up notation and prove some basic results on weak multiplicity of (not necessarily commuting) n-tuples of operators on Hilbert spaces. In §3, we study a lower bound multiplicity of joint invariant subspaces

of a class of commuting n-tuples of operators. The main theorem on additivity formula is proved in§4. The paper is concluded in§5 with corollaries of the main theorem and some general discussions.

2. Notation and basic results

In this section, we introduce the notion of weak multiplicities and describe some prepara- tory results. This notion is not absolutely needed for the main results of this paper as we shall mostly work in the setting of multiplicities. However, we believe that the idea of weak multiplicities of (not necessary commuting) tuples of operators might be of inde- pendent interest. Throughout this paper, the following notation will be adopted: T_i is a bounded linear operator on a separable Hilbert space H_i,i= 1, . . . , n, and n≥2. We set

H˜ =H1⊗ · · · ⊗ Hn

and

T˜= ( ˜T₁, . . . ,T˜_n).

where

T˜_i=I_H1⊗ · · · ⊗I_Hi−1⊗T_i⊗I_Hi+1⊗ · · · ⊗I_Hn∈ B( ˜H),

for all i= 1, . . . , n. It is now clear that ( ˜T₁, . . . ,T˜_n) is a doubly commuting tuple of operators on ˜H(that is, ˜T_iT˜_j= ˜T_jT˜_i and ˜T_p^∗T˜_q = ˜T_qT˜_p^∗ for all 1≤i, j≤nand 1≤p <

q≤n). Moreover, if mult_Ti(Hi) = 1 for alli= 1, . . . , n, then mult_T˜( ˜H) = 1. We denote byDⁿ the unit polydisc inCⁿ and byz the element (z₁, . . . , z_n) inCⁿ.

The above notion of ‘tensor product of operators’ is suggested by natural (and analytic) examples of reproducing kernel Hilbert spaces over product domains inCⁿ. For instance, if{α₁, . . . , α_n} ⊆N, then

K_α(z,w) :=

n i=1

1

(1−z_iw¯_i)^αi (z,w ∈Dⁿ),

is a positive definite kernel overDⁿ, and the multiplication operator tuple (M_z1, . . . , M_zn) defines bounded linear operators on the corresponding reproducing kernel Hilbert space L²_α(Dⁿ) (known as the weighted Bergman space overDⁿ with weightα= (α₁, . . . , α_n)).

It follows that (cf. [19])

H˜=L²_α1(D)⊗ · · · ⊗L²_αn(D), and M˜_z= ( ˜M_z1, . . . ,M˜_zn),

whereM_zi denotes the multiplication operatorM_zonL²_αi(D),i= 1, . . . , n. In particular, ifα= (1, . . . ,1), then ˜H=H²(Dⁿ) is the well-known Hardy space over the unit polydisc.

We also refer the reader to Popescu [14,15] for elegant and rich theory of ‘tensor product of operators’ in multivariable operator theory.

Let H be a Hilbert space, and let A= (A₁, . . . , A_n) be an n-tuple (not necessarily commuting) of bounded linear operators onH. Let

w-mult_A(H) = min{#G: [G]_A=H, G⊆ H}, where

[G]_A= span{A^k(G) :k∈Zⁿ₊},

andA^k=A^k₁¹· · ·A^k_nⁿfor allk ∈Zⁿ₊. If w-mult_A(H) =m <∞, then we say that theweak multiplicity ofAism. We say that Aisweakly cyclic if w-mult_A(H) = 1. A subset Gof His said to be weakly generating with respect toA if [G]_A=H.

Now letLbe a closed subspace of H. Then WA(L) :=L

n i=1

A_iL,

is called the wandering subspaceofL with respect toP_LA|L. If, in addition L=

k∈Zⁿ₊

(P_LA|_L)^k(W_A(L)),

then we say thatP_LA|Lsatisfies theweakly generating wandering subspace property. Here P_LA|L = (P_LA₁|L, . . . , P_LA_n|L) and

(P_LA|L)^k = (P_LA₁|L)^k1· · ·(P_LA_n|L)^kn, for allk ∈Zⁿ₊.

Note that if A is commuting and L is joint A-invariant subspace (that is, A_iL ⊆ L for all i= 1. . . , n), then weakly generating wandering subspace property is commonly known asgenerating wandering subspace property.

We now proceed to relate weak multiplicities and dimensions of weakly generating wandering subspaces. LetAbe ann-tuple of bounded linear operators onH,Lbe a joint A-invariant subspace ofH, and letMbe a closed subspace of L. Then

P_WA(L)([M]_A) =P_WA(L)(M), since

P_WA(L)(A^kM) = 0 for allk ∈Zⁿ₊\ {0}.

Now suppose that [M]_A=L, that is, M is a weakly generating subspace of L with respect to A. Then

W_A(L) =P_WA(L)(M).

Hence

w-mult_A|L(L)≥dimWA(L).

Moreover, ifLsatisfies the weakly generating wandering subspace property, then w-mult_A(L) = dimWA(L).

Therefore, we have proved the following:

Proposition 2.1. LetLbe a closed jointA-invariant subspace ofH. IfLsatisfies the weakly generating wandering subspace property with respect toA_L, then w-mult_A(L) = dimW_A(L).

We now proceed to a variation of Lemma 2.1 in [5] which relates the multiplicity of a commuting tuple of operators with the weak-multiplicity of the compressed tuple to a semi-invariant subspace.

Lemma 2.2. LetA be ann-tuple of bounded linear operators on a Hilbert spaceH.

Let L1 and L2 be two joint A-invariant subspaces of H and L2⊆ L1. If L=L1 L2, then

w-mult_PLA|L(L)≤w-mult_A|L

1(L₁).

Proof. We have P_LA_jP_L =P_LA_jP_L1−P_LA_jP_L2 and thus by A_jL2⊆ L2 we infer that

P_LA_jP_L=P_LA_jP_L1, for allj= 1, . . . , n. Since A_jL1⊆ L1, we have

(P_LA_iP_L)(P_LA_jP_L) =P_LA_iP_L1A_jP_L1, that is

(P_LA_iP_L)(P_LA_jP_L) =P_L(A_iA_j)P_L1, for alli, j= 1, . . . , n, and so

(P_LAP_L)^k=P_LA^kP_L1,

for all k ∈Zⁿ₊. Clearly, ifGis a minimal generating subset of L1 with respect toA|L1, thenP_LGis a generating subset ofLwith respect toP_LA|L, and thus w-mult_PLA|L(L)≤ w-mult_A|L

1(L1). This completes the proof of the lemma.

In particular, if L₁=H, thenQ:=H L₂ is a joint (A^∗₁, . . . , A^∗_n)-invariant subspace ofH. In this case, denote byC_i=P_QA_i|_Qthe compression ofA_i,i= 1, . . . , n, and define then-tuple onQas

C_Q= (C₁, . . . , C_n).

Then we have the following estimate:

w-mult_CQ(Q)≤w-mult_A(H).

Moreover, we also have

Corollary 2.3. Let A= (A₁, . . . , A_n) be a commuting tuple of bounded linear operators on a Hilbert spaceH. IfQis a closed joint A^∗-invariant subspace ofH, then

mult_CQ(Q)≤mult_A(H).

This has the following immediate (and well-known) application: Suppose A is a commuting tuple onH. IfAis cyclic, thenC_QonQis also cyclic.

3. A lower bound for multiplicities

In this section, we first lay out the setting of joint invariant subspaces of our discussions throughout the paper. Then we present a lower bound of multiplicities of those joint invariant subspaces. We begin by recalling the following useful lemma (cf. Lemma 2.5, [18]):

Lemma 3.1. If {A_i}ⁿ_i=1 is a commuting set of orthogonal projections on a Hilbert spaceK, thenL=_n

i=1ranA_i is a closed subspace ofK, and

P_L=I−ⁿ

i=1

(I−A_i)

=A₁(I−A₂). . .(I−A_n)⊕A₂(I−A₃). . .(I−A_n)⊕. . .+A_n−1(I−A_n)⊕A_n. Next, we introduce the invariant subspaces of interest. Again, we continue to follow the notation as introduced in§2.

LetHibe a Hilbert space,T_i a bounded linear operator onHi, and letQi be a closed T_i^∗-invariant subspace ofHi,i= 1, . . . , n. SetSi=Q^⊥_i and

P_i=P_Si and Q_i=I_Hi−P_Si,

for alli= 1, . . . , n. Recall again that

P˜_i =I_H1⊗. . .⊗I_Hi−1⊗P_Si⊗I_Hi+1⊗. . .⊗I_Hn∈ B( ˜H), and

P˜_iP˜_j = ˜P_jP˜_i, for alli, j= 1, . . . , n. By Lemma3.1, it then follows that

S= n i=1

ran ˜P_i, (3.1)

is a joint ˜T-invariant subspace of ˜H. Moreover,

S= (Q₁⊗ · · · ⊗ Q_n)^⊥.

Our main goal is to compute the multiplicity of the commuting tuple T˜|_S = ( ˜T₁|_S, . . . ,T˜_n|_S) onS.

For each i= 1, . . . , n, define X_i∈ B( ˜H) by

X_i = ˜P_iQ˜_i+1. . .Q˜_n. ThenX_i²=X_i=X_i^∗ and

X_pX_q = 0,

for alli= 1, . . . , n, andp=q. This implies that{X_i}ⁿ_i=1is a set of orthogonal projections with orthogonal ranges. Then, by virtue of (3.1), one can further rewriteS as

S= n i=1

ran ˜P_i = n

i=1

ranX_i, (3.2)

and by Lemma3.1, one representsP_S as P_S =

n i=1

X_i. Define

F = ranX₁⊕ran( ˜Q₁X₂)⊕ · · · ⊕ran( ˜Q₁· · ·Q˜_n−1X_n). (3.3) Then, as easily seen

Q˜_iX_j=X_jQ˜_i, for all 1≤i≤j andj = 1, . . . , n, it follows that

ran( ˜Q₁· · ·Q˜_pX_p+1)⊆ranX_p+1, for allp= 1, . . . , n−1, and consequently

S ⊇ F.

Our first aim is to analyse the closed subspace F and to construct n−1 nested (and suitable) closed subspaces{F_i}ⁿ⁻¹_i=1 such that

S ⊇ F₁⊇ · · · ⊇ F_n−1=F. To this end, first set

F₁= ranX₁⊕ranX₂⊕ · · · ⊕ranX_n−1⊕ran( ˜Q_n−1X_n), and define

F2= ranX₁⊕ran( ˜Q₁X₂)⊕ · · · ⊕ran( ˜Q₁X_n−1)⊕ran( ˜Q₁Q˜_n−1X_n).

We then proceed to define Fi, i= 2, . . . , n−1, as F_i= ran

X₁⊕Q˜₁X₂⊕ · · · ⊕ _i−1

t=1

Q˜_t

X_i⊕ · · · ⊕ _i−1

t=1

Q˜_t

X_n−1

⊕ _i−1

t=1

Q˜_tQ˜_n−1

X_n

.

Therefore

P_Fi=X₁⊕Q˜₁X₂⊕ · · · ⊕ _i−1

t=1

Q˜_t

X_i⊕ · · · ⊕ _i−1

t=1

Q˜_t

X_n−1⊕ _i−1

t=1

Q˜_tQ˜_n−1

X_n, (3.4) for alli= 2, . . . , n−1. Therefore, denoting

A= _i−2

t=1

Q˜_t

P˜_i−1, we have

P_Fi−1Fi =A(X_i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n), (3.5) for alli= 2, . . . , n−1. SinceAX_p=X_pA for allp=i, . . . , n, the above formula yields

P_Fi−1Fi = (X_i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n)A.

Let i∈ {2, . . . , n−1} be a fixed natural number. We claim that F_i−1 F_i is a joint P_Fi−1T P˜ _Fi−1-invariant subspace, that is

P_Fi−1T˜_j(F_i−1 F_i)⊆ F_i−1 F_i. or, equivalently

(P_Fi−1T˜_jP_Fi−1)P_Fi−1Fi=P_Fi−1FiT˜_j|_Fi−1Fi, for allj= 1, . . . , n. There are four cases:

Case I: Ifj > i, then one has ˜T_jA=AT˜_j and so

P_Fi−1FiT˜_jP_Fi−1Fi =A(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n) ˜T_j(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n).

On the other hand, since

P_Fi−1T˜_jP_Fi−1Fi =P_Fi−1AT˜_j(X_i⊕ · · · ⊕X_j⊕ · · · ⊕Q˜_n−1X_n), and

P_Fi−1 =X₁⊕( ˜Q₁X₂)⊕ · · · ⊕ _i−2

t=1

Q˜_tX_i−1

⊕ _i−2

t=1

Q˜_tX_i

⊕ · · · ⊕ _i−2

t=1

Q˜_tX_n−1

⊕ _i−2

t=1

Q˜_tQ˜_n−1X_n

,

it follows that

P_Fi−1T˜_jP_Fi−1Fi =A(X_i−1⊕X_i⊕ · · · ⊕Q˜_n−1X_n) ˜T_j(X_i⊕ · · · ⊕Q˜_n−1X_n), as X_tA= 0 for allt= 1, . . . , i−2, and ⁱ⁻²_t=1Q˜_tA=A. Moreover, since

X_i−1T˜_j= ( ˜P_i−1Q˜_i· · ·Q˜_j· · ·Q˜_n) ˜T_j= ˜P_i−1Q˜_i· · ·Q_jT_jQ_j· · ·Q˜_n,

it follows that

X_i−1T˜_jX_t= 0, for allt=i, . . . , n. This leads to

P_Fi−1T˜_jP_Fi−1Fi=A(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n) ˜T_j(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n).

Case II:Ifj=i, then

T˜_iP_Fi−1Fi=A((T_iP_iQ˜_i+1· · ·Q˜_n)⊕T˜_iX_i+1⊕ · · · ⊕T˜_iQ˜_n−1X_n), implies that

P_Fi−1T˜_iP_Fi−1Fi= _i−2

t=1

Q˜_t

(X_i−1⊕X_i⊕ · · · ⊕Q˜_n−1X_n) ˜T_iP_Fi−1Fi

= _i−2

t=1

Q˜_t

(X_i−1⊕X_i⊕ · · · ⊕Q˜_n−1X_n) ˜T_iA

×(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n)

=A(X_i−1⊕X_i⊕ · · · ⊕Q˜_n−1X_n) ˜T_i(X_i⊕X_i+1⊕ · · · ⊕Q˜_n−1X_n)

=P_Fi−1FiT˜_iP_Fi−1Fi,

where the next-to-last equality follows from the fact again thatAT˜_i= ˜T_iA, ( ⁱ⁻²_t=1Q˜_t)A= Aand X_i−1T˜_iX_t= 0 for allt=i, . . . , n.

Case III:Letj =i−1. Since T˜_i−1A=

_i−2

t=1

Q˜_t

T_i−1P_i−1=AT_i−1P_i−1,

by setting

Aˆ= _i−2

t=1

Q˜_t

T_i−1P_i−1,

it follows that

T˜_i−1P_Fi−1Fi= ˆAX_i⊕AXˆ _i+1⊕ · · · ⊕AXˆ _n−1⊕AˆQ˜_n−1X_n. ThenX_pAˆ= ˆAX_p for allp=i, . . . , n, andAAˆ= ˆAimplies that

P_Fi−1T˜_i−1P_Fi−1Fi = ˆA(X_i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n)

=P_Fi−1FiT˜_i−1P_Fi−1Fi,

where the second equality follows from (3.5) and the fact thatT_i−1P_i−1=P_i−1T_i−1P_i−1.

Case IV: Letj < i−1. Then it is clear that

T˜_jP_Fi−1Fi= ˆA(X_i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n),

where ˆA= ˜T_jA, that is

Aˆ= ˜Q₁· · ·Q˜_j−1T_jQ_jQ˜_j+1· · ·Q˜_i−2P˜_i−1.

Note thatX_tAˆ= ˆAX_t for allt=i, . . . , n, and

AAˆ= ˜Q₁· · ·Q˜_j−1Q_jT_jQ_jQ˜_j+1· · ·Q˜_i−2P˜_i−1.

SinceX_pX_q =δ_pqX_p for allpandq, it follows that

P_Fi−1T˜_jP_Fi−1Fi=AA(Xˆ _i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n).

On the other hand, the representation of ˜T_jP_Fi−1Fi above and (3.5) yields P_Fi−1FiT˜_jP_Fi−1Fi =AA(Xˆ _i⊕X_i+1⊕ · · · ⊕X_n−1⊕Q˜_n−1X_n) and proves the claim.

We turn now to prove that (P_FiT˜₁|_Fi, . . . , P_FiT˜_n|_Fi) is a commuting tuple for all i= 1, . . . , n−1, that is

P_FiT˜_sP_FiT˜_tP_Fi =P_FiT˜_tP_FiT˜_sP_Fi,

for alls, t= 1, . . . , n. Fix ani∈ {1, . . . , n−1}and let

P_Fi=M₁⊕ · · · ⊕M_n, (3.6) where M_j,j= 1, . . . , n, denotes thejth summand in the representation ofP_Fi in (3.4).

Recalling the terms in (3.4), we see thatM_j is a product ofndistinct commuting orthog- onal projections of the form ˜P_k, ˜Q_l and ˜I_Hm, 1≤k, l, m≤n. For eachs= 1, . . . , n, we set

M_j =M_j,sMˆ_j,s,

where M_j,s is thesth factor ofM_j and ˆM_j,s is the product of the same factors ofM_j, except the sth factor of M_j is replaced by ˜I_Hs. Note again thatM_j,s= ˜P_s,Q˜_s, or ˜I_Hs.

We first claim that

M_jT˜_sM_k = 0, (3.7)

for allj=k. Indeed, ifM_j,s= ˜Q_s, thenM_jT˜_sM_k=M_j,sMˆ_j,sT˜_sM_k yields M_jT˜_sM_k =M_j,sT˜_sMˆ_j,sM_k=M_j,sT˜_sM_j,sMˆ_j,sM_k=M_j,sT˜_sM_jM_k= 0, as ˜Q_sT˜_sQ˜_s= ˜Q_sT˜_s. Similarly, if M_j,s= ˜P_s, then

M_jT˜_sM_k =M_jMˆ_k,sT˜_sM_k,s=M_jMˆ_k,sM_k,sT˜_sM_k,s=M_jM_kT˜_sM_k,s= 0, as ˜P_sT˜_sP˜_s= ˜T_sP˜_s. The remaining case,M_j,s= ˜I_Hs, follows from the fact that

M_jT˜_sM_k= ˜T_sM_jM_k.

This proves the claim. Hence the representation ofP_FiT˜_sP_Fi simplifies as

P_FiT˜_sP_Fi =M₁T˜_sM₁⊕ · · · ⊕M_nT˜_sM_n. (3.8) Thus,

P_FiT˜_sP_FiT˜_tP_Fi =M₁T˜_sM₁T˜_tM₁⊕ · · · ⊕M_nT˜_sM_nT˜_tM_n. Now ifs=t, then for eachj= 1, . . . , n, we have

M_jT˜_sM_jT˜_tM_j =M_jMˆ_j,sT˜_sM_j,sM_j,tT˜_tMˆ_j,tM_j

= (M_jMˆ_j,sM_j,t) ˜T_sT˜_t(M_j,sMˆ_j,tM_j)

=M_jT˜_sT˜_tM_j, and hence

(P_FiT˜_sP_Fi)(P_FiT˜_tP_Fi) =M₁T˜_sT˜_tM₁⊕ · · · ⊕M_nT˜_sT˜_tM_n.

This completes the proof of the commutativity property of the tuple (P_FiT˜₁|_Fi, . . . , P_FiT˜_n|_Fi),i= 1, . . . , n−1. Furthermore, ifs=t, then

(M_jT˜_sM_j)²=M_jT˜_s²M_j. Indeed, if M_j,s= ˜Q_s, thenM_jT˜_sM_j=M_jT˜_sMˆ_j,s gives us

M_jT˜_sM_jT˜_sM_j=M_jT˜_sMˆ_j,sT˜_sMˆ_j,sM_j =M_jT˜_sT˜_sMˆ_j,sM_j =M_jT˜_s²M_j. Similarly, ifM_j,s= ˜P_s or ˜I_Hs, thenM_jT˜_sM_j= ˜T_sM_j, and hence

M_jT˜_sM_jT˜_sM_j=M_jT˜_s²M_j. Hence we obtain

(P_FiT˜_sP_Fi)(P_FiT˜_tP_Fi) =M₁T˜_sT˜_tM₁⊕ · · · ⊕M_nT˜_sT˜_tM_n, (3.9) for alls, t= 1, . . . , n.

Therefore, with the notation introduced above, we have proved the following: