Tensor algebras, displacement structure, and the Schur algorithm

17 Pages·0.21 MB·English

by  T. Constantinescu

TENSOR ALGEBRAS, DISPLACEMENT STRUCTURE,
AND THE SCHUR ALGORITHM
T. CONSTANTINESCU AND J. L. JOHNSON
1
0
0 Abstract. In this paper we explore the connection between tensor algebras and
2
displacement structure. We describe a scattering experiment in this framework, we

n obtaina realizationof the elements ofthe tensor algebraas tranfer maps ofa certain
a
class ofnonstationarylinear systems,andwe describe a Schur type algortihmfor the
J
Schur elements of the tensor algebra.
0
3

]
A
F 1. Introduction
.
h
It was recently showed in [5] that the tensor algebras have displacement structure.
t
a
Thegoalofthispaperistofurther developthisconnection. The displacement structure
m
theory, as it was initiated in the paper [8], consists in recursive factorizationof matrices
[
with additional structure encoded by a so-called displacement equation. Many applica-
1
tions of this theory were found in applied fields, like in the study of wave propagation
v
2 in layered media, or filtering and modeling of nonstationary processes (see [9] for a
5
survey). On the mathematical side, there are applications to bounded interpolation in
2
1 upper triangular algebras (including the classical setting of bounded analytic functions
0 on the unit disk) and to triangular factorizations of operators.
1
0 On the other hand, the tensor algebra is a classical concept and in [5] it is showed
h/ that there is a connection expressed by the fact that any element T of the unit ball of
t the algebra (with respect to an appropriate operator norm) satisfies the equation
a
m
N
v: A− F AF∗ = GJG∗,
k k
i X
X k=1
r where A = I − T∗T and the so-called generators F , G and J are described in the
a k
next section. This equation is of displacement type so we can develop a theory for the
tensor algebra that parallels the one dimensional case. It is the main goal of this paper
to describe several such results.
The paper is organized as follows. In Section 2 we review the basic elements relevant
to the tensor algebra and to the so-called noncommutative analytic Toeplitz algebra
which is a natural extension of the tensor algebra introduced in [11], and studied in
other papers such as [2], [6], [11], [12]. We describe an isometric isomorphic represen-
tation of this later algebra by a certain algebra U (H) of upper triangular bounded
T
operators and this represention allows us to rely on the results and ideas in the dis-
placement structure theory. In this section we also show that a certain class of positive
1
definite kernels on the free semigroup with N generators, introduced in [12], also has
displacement structure of the type mentioned above and in Corollary 2.5 we explain
the connection between this class of positive definite kernels and the algebra U (H)
T
which is well-known for N = 1. In Section 3 we obtain results corresponding to so-
called scattering experiments. Theorem 3.1 is the main result in this direction and as
a by-product we can obtain Schur type parametrizations for U (H) and the realization
T
of the elements of the algebra as transfer maps of a certain class of nonstationary linear
systems. Section 4 develops a Schur type algorithm for U (H). We plan to explore ap-
T
proximation properties (Szeg¨o type theory) related to this algorithm in a future paper.
This paper is part of [7].
2. Preliminaries
We briefly review the construction of the tensor algebras and we explain their dis-
placement structure.
2.1. Tensor Algebras. We introduce some notation relevant to tensor algebras and
we describe their representation as algebras of upper triangular operators (see, for
instance, [10]). The associative tensor algebra T(H) generated by the complex vector
space H is defined by the algebraic direct sum
T (H) = ⊕ H⊗k,
k≥0
where
H⊗k = H⊗...⊗H
k factors
is the k-fold algebraic tensor product o|f H {wzith i}tself. The elements of T(H) are
terminating sequences x = (x ,x ,... ,x ,0,...), where x ∈ H⊗p is called the pth
0 1 k p
homogeneous component of x (note that H⊗0 = C, H⊗1 = H). The addition and
multiplication of elements in T (H) are defined componentwise:
(x+y) = x +y
n n n
and
(xy) = x ⊗y , (x ⊗y = y ⊗x = x y ).
n k l 0 n n 0 0 n
X
k+l=n
In order to deal with displacement structure matters it is convenient to use the fact
that T(H) can be represented by upper triangular matrices of a special type. For a
certain simplicity, from now on we take H = CN, N ≥ 1. Then we denote by F(H)
the full Fock space associated to H, that is, the Hilbert space
F(H) = ⊕ H⊗k
k≥0
obtained by taking the Hilbert direct sum of the spaces H⊗k, k ≥ 0, on which we
consider the tensor Hilbert space structure induced by the Euclidean norm on H =
CN. If {e ,e ,... ,e } is the standard basis of CN, then {e ⊗ ...e | i ,... ,i ∈
1 2 N i1 ik 1 k
{1,2,... ,N}} is an orthonormal basis for H⊗k. It is more convenient to write e
σ
insteadofe ⊗...e ,whereσ = i ...i isviewed asawordintheunitalfreesemigroup
i1 ik 1 k
2
F+ with N generators 1, ... ,N. The empty word is the identity element of F+. The
N N
length of the word σ is denoted by |σ|. We also use to write the elements of F+
N
in lexicographic order. The space H⊗k+1 can be identified with the direct sum of N
copies of H⊗k. More precisely, we notice that {(δ e )N | j ∈ {1,... ,N},|σ| = k} is
jl σ l=1
an orthonormal basis for H⊗k ⊕...⊕H⊗k = (H⊗k)⊕N (δ is the Kronecker symbol).
jl
N terms
We deduce that the mapping
| {z }
φ (((δ e )N )) = e , j = 1,... ,N,
k jl σ l=1 jσ
extends to a unitary operator from (H⊗k)⊕N onto H⊗k+1 and the formula
ψ = φ φ⊕N ...φ⊕Nk−2, k ≥ 1,
k k−1 k−2 1
gives a unitary operator from H = H⊕Nk−1 onto H⊗k (the notation T⊕l means the
k
direct sum of l copies of the operator T). We finally obtain a unitary operator ψ :
⊕∞ H → F(H) by the formula ψ = ⊕∞ ψ , where ψ is the identity on C = H .
k=0 k k=0 k 0 0
Each H , k ≥ 2, is also identified with the direct sum of N copies of H ,
k k−1
H = H⊕Nk−2 ⊕...⊕H⊕Nk−2,
k
N terms
| {z }
but the explicit mention of the identification can be omitted in this case.
An operator T ∈ L(⊕∞ H ) has a matrix representation T = [T ]∞ with T ∈
k=0 k ij i,j=0 ij
L(H ,H ). We say that T is upper triangular provided that T = 0 for i > j. We
j i ij
denote by U0(H) the set of upper triangular operators T = [T ]∞ in L(⊕∞ H )
T ij i,j=0 k=0 k
with the property that for i ≤ j and i,j ≥ 1,
(2.1) T = T⊕N
ij i−1,j−1
and T = 0 for j > k and some k. We notice that the entries T , j ≥ 0, determine
0j 0j
the matrix T. Also, a simple calculation shows that U0(H) is an associative algebra.
T
We define an algebra isomorphism
(2.2) Φ : T(H) → U0(H)
T
as follows: for x ∈ T(H), x = (x ,x ,...), there are complex numbers c , σ ∈ F+,
0 1 σ N
such that
(2.3) x = c e .
j σ σ
X
|σ|=j
For j ≥ 0, T is the row matrix [c ] . Then T = 0 for sufficiently large j and we
0j σ |σ|=j 0j
can define T ∈ L(⊕∞ H ) by using (2.1). Set Φ(x) = T. Clearly Φ is a bijection.
k=0 k
In order to see that Φ is also an algebra morphism it is convenient to introduce the
following operators. First, the operator C−, j = 1,... ,N, is an element of U0(H)
j T
defined by the formula: C− = Φ(0,e ,0,...) = [Tj]∞ , where Tj = 0 for k 6= 1 and
j j lk l,k=0 0k
Tj = e . Then C+ = (C−)∗, j = 1,... ,N. If we use the notation C− to denote the
01 j j j σ
3
operator C−...C−, where σ = i ...i , then we deduce that each T ∈ U0(H) has a
i1 ik 1 k T
representation
(2.4) T = c C−.
σ σ
σX∈F+
N
Now the required properties of Φ follow from the easily verifiable fact that
Φ((0,e ,0,...)⊗(0,e ,0,...)) = C−C−.
j k j k
It appears to be quite natural to consider the algebra U (H) of all those opera-
T
tors in L(⊕∞ H ) that satisfy the condition (2.1). It turns out that U (H) is iso-
k=0 k T
metrically isomorphic to the so-called non-commutative analytic Toeplitz algebra, re-
cently studied in papers like [11], [2], [6]. We also denote by S(H) the Schur class
of all contractions in U (H). We can extend this setting as folows. Let E and E
T 1 2
be Hilbert spaces, then U (H,E ,E ) denotes the set of the operators T = [T ]∞
T 1 2 ij i,j=0
in L(⊕∞ (H ⊗ E ),⊕∞ (H ⊗ E )) obeying (2.1). Instead of U (H,E,E) we write
k=0 k 1 k=0 k 2 T
U (H,E). Also S(H,E ,E ) denotes the set of the contractions in U (H,E ,E ). We
T 1 2 T 1 2
notice that U (H) = U (H,C,C).
T T
2.2. Displacement Structure. The systematic study of the displacement structure
was initiated in [8], and since then manifold applications have been found in the fields
of mathematics and electrical engineering (for a recent surveys, see [9]). Two main
themes of this theory are about scattering experiments and recursive factorizations. A
matrix R has displacement structure with respect to the generators F and G provided
that
(2.5) A−FAF∗ = GJ G∗,
pq
where J = I ⊕−I is a signature matrix that specifies the displacement inertia of A.
pq p q
In many applications, r = p+q is much smaller than the size of A. From now on we
will write J instead of J . There is also a version of this theory allowing A to depend
11
on a (usually integer) paramater, see [9] or [4]. A major result concerning matrices
with displacement structure is that, under suitable assumptions on the generators, the
succesive Schur complements of A inherit a similar structure. This allows to write the
Gaussian elimination for A only in terms of generators (see [8], [9]).
We now describe a connection between displacement structure and U (H). Let
T
C = [Tk]∞ , k = 1,... ,N, be isometries defined by the formulae: T = 0 for
k ij i,j=0 ij
i 6= j + 1 and T is a block-column matrix consisting of N blocks of dimension
i+1,i
dimH , all of them zero except for the kth block which is I . These operators are
i Hi
closely related to the operators C+, in the sense that there is a unitary operator u such
k
that C = uC+u∗, k = 1,... ,N. The following result was noticed in [5].
k k
Theorem 2.1. Let T = [T ]∞ ∈ S(H) and A = I −T∗T. Then
ij i,j=0
N
(2.6) A− C AC∗ = GJG∗,
k k
X
k=1
4
where the generator G is given by the formula:
1 T∗
00
G =  0 T0∗1 .
. .
. .
. .
 
The equation (2.6) can be rewritten in the form of a time variant displacement
equation by setting F = [C ,...C ]. We now discuss some additional examples.
1 N
2.2.1. Positive definite kernels on F+. Amapping K : F+×F+ → Ciscalledapositive
N N N
definite kernel on F+ provided that
N
k
K(σ ,σ )λ λ ≥ 0
i j j i
iX,j=1
for each positive integer k and each choice of words σ ,... ,σ in F+ and complex
1 k N
numbers λ ,... ,λ . Without lossofgenerality, we assume K(∅,∅) = 1. The semigroup
1 k
F+ acts on itself by juxtaposition and we assume that the positive definite kernel K is
N
invariant under this action, that is
K(τσ,τσ′) = K(σ,σ′), τ,σ,σ′ ∈ F+.
N
By the invariant Kolmogorov decomposition theorem, see e.g., [10], Ch. II, there exists
an isometric representation u of F+ on a Hilbert space K and a mapping v : F+ → K
N N
such that
K(σ,τ) = hv(τ),v(σ)i ,
K
u(τ)v(σ) = v(τσ)
for all σ,τ ∈ F+, and the set {v(σ) | σ ∈ F+} is total in K. The mapping v is unique
N N
up to unitary equivalence and u is uniquely determined by v. Also, the previous
representation of K implies, as noticed in [12], that u(1), ..., u(N) are isometries with
orthogonal ranges if and only if K(σ,τ) = 0 for all pairs (σ,τ) with the property that
there is no α ∈ F+ such that σ = ατ or τ = ασ. In this case, K is called in [12] a
N
multi-Toeplitz kernel and we will use the same terminology. We can define
(2.7) S = [K(σ,τ)] ∈ L(H ,H ).
ij |σ|=i,|τ|=j j i
By induction on the length of words and using properties of the lexicographic order
on words we check that if K is a positive definite multi-Toeplitz kernel, then for all
i,j ≥ 1,
(2.8) S = S⊕N .
ij i−1,j−1
This relation is an analogue of (2.1), therefore we expect that positive definite multi-
Toeplitz kernels have displacement structure.
5
Theorem 2.2. Let S = [S ]∞ be the array of operators associated to a positive
ij i,j=0
definite multi-Toeplitz kernel on F+. Then
N
N
(2.9) S − C SC∗ = GJG∗,
k k
X
k=1
where the generator G is given by the formula:
1 0
G =  S0∗1 S0∗1 .
. .
. .
. .
 
The equality above can be interpreted in the sense of an equality of unbounded oper-
ators with dense domain made of the terminating sequences in ⊕∞ H .
l=0 l
Proof. We write S = L+Q, where L = [L ]∞ and Q = [Q ]∞ are the lower and,
ij i,j=0 ij i,j=0
respectively, the upper parts of S, such that L = S for i > j and L = 0 for i ≤ j,
ij ij ij
while Q = S for i ≤ j and Q = 0 for i > j. Since, formally (that is, computing the
ij ij ij
entries of the arrays that are involved according to the usual rule of multiplication of
operators), LC = C L and QC∗ = C∗Q for all k = 1,... ,N, we deduce that
k k k k
S − N C SC∗ = L+Q− N C (L+Q)C∗
k=1 k k k=1 k k
P P
= L+Q− N C LC∗ − N C QC∗
k=1 k k k=1 k k
P P
= L+Q− N LC C∗ − N C C∗Q.
k=1 k k k=1 k k
P P
But I = N C C∗ = P , the orthogonal projection on H , so we can deduce that
k=1 k k H0 0
P
S − N C SC∗ = L+Q−L(I −P )−(I −P )Q
k=1 k k H0 H0
P
= LP +P Q = GJG∗.
H0 H0
We now consider truncations of a positive definite multi-Toeplitz kernel on F+. If K
N
is such a kernel and S are associated to K by the formula (2.7), then we define K =
ij n
[S ]n . These are positive operators on ⊕n H and their entries satisfy (2.8). We
ij i,j=0 l=0 l
also introduce the operators C = P C | ⊕n H . We obtain another example
k,n ⊕nl=0Hl k l=0 l
of a displacement equation.
Corollary 2.3. The positive operator K satisfies the displacement equation
n
N
(2.10) K − C K C∗ = GJG∗,
n k,n n k,n
X
k=1
6
where the generator G is given by the formula:
1 0
S∗ S∗
 01 01 
G = . . .
. .
. .
 
 S∗ S∗ 
 0n 0n 
Proof. This is an immediate consequence of Theorem 2.2.
It turns out that the operators satisfying (2.10) have a certain explicit description.
For x ∈ ⊕n H we denote by U(x) the upper triangular operator on ⊕n H satisfying
l=0 l l=0 l
(2.8) and whose first row is x.
Theorem 2.4. A positive operator A on ⊕n H satisfies (2.10) if and only if A =
l=0 l
U(x)∗U(x) −U(y)∗U(y) for two vectors x,y ∈ ⊕n H with the property that there is
l=0 l
another vector z ∈ ⊕n H such that U(y) = U(z)U(x) and U(z) is a contraction.
l=0 l
Proof. If A = U(x)∗U(x) − U(y)∗U(y) then we use the fact that for any x ∈ ⊕n H
l=0 l
and k = 1,... ,N, C U(x)∗ = U(x)∗C , and deduce (2.10) as in the proof of Theo-
n,k n,k
rem 2.2. Conversely, if A satisfies (2.10), then we use the fact that C = 0 for |σ| > n
σ,n
in order to deduce that
n
A = GJG∗ + C GJG∗C∗ .
σ,n σ,n
X
|σ|=1
Hence, if G = [x∗,y∗] for some x,y ∈ ⊕n H , then A = U(x)∗U(x)−U(y)∗U(y). Since
l=0 l
A is positive, it follows that there exists a contraction T such that U(y) = TU(x).
We notice that we can assume, without loss of generality, that x 6= 0 and then we
0
deduce that C T∗ = T∗C for k = 1,... ,N, which implies that T = U(z) for some
n,k n,k
z ∈ ⊕n H .
l=0 l
We mention a simple but useful consequence of this result. Let K be a positive
definite multi-Toeplitz kernel on F+ with K(∅,∅) = 1 and let S , i,j ≥ 0, be the
N ij
operators associated to K by (2.7). Let
1 0
S∗ S∗
 01   01 
x = . , y = . ,
n . n .
. .
   
 S∗   S∗ 
 0n   0n 
then K = U(x )∗U(x ) − U(y )∗U(y ) and by the previous result there exists z ∈
n n n n n n
⊕n H such that U(y ) = U(z )U(x ). In fact, z is uniquely determined and also
l=0 l n n n n
P U(z ) | ⊕n H = U(z ) for all n ≥ 1. It follows that there exists a unique
⊕nl=0Hl n+1 l=0 l n
T(K) ∈ S(H) such that P T | ⊕n H = U(z ) and T(K) = 0. Denote by S (H)
⊕nl=0Hl l=0 l n 00 0
the set of those elements T ∈ S(H) with T = 0.
00
Corollary 2.5. The mapping K → T(K) gives a one-to-one correspondence between
the set of positive definite multi-Toeplitz kernels on F+ with K(∅,∅) = 1 and S (H).
N 0
7
2.2.2. Pick kernels. This type of kernels were recently introduced in [2], [12], and they
were shown to have a certain universality property with respect to a Nevanlinna-Pick
problem, [1].
Let BN be the open unit ball in CN. For distinct points λ , ..., λ in BN and
1 L
complex numbers b , ..., b , we define the Pick kernel by the formula
1 L
L
1−b b
R = j l ,
(cid:20)1−hλ ,λ i(cid:21)
j l j,l=1
where hλ ,λ i denotes the Euclidean inner product in CN.
j l
We can define
F = diag(λ(j))L , j = 1,... ,N,
j l l=1
the diagonal matrix with the diagonal made of the jth components of the points λ ,
1
..., λ . Also, we set
L
1 b
1
1 b
 2 
G = . . .
. .
. .
 
 1 b 
 L 
We can check by direct computation that
N
(2.11) R− F RF∗ = GJG∗.
k k
X
k=1
It was shown in [5] how to use this equation in order to solve some multidimensional
Nevanlinna-Pick type problems as those in [2], [12].
3. Scattering Experiments
One of the main results in displacement structure theory refers to the possibility of
associating a scattering operator to the data given by the generators of the equation
(2.5). In this section we prove a similar result for displacement equations of the type of
(2.11) and we explain the connection with a special class of time variant linear systems.
Let E , E , and G be Hilbert spaces. We consider a displacement equation of the
1 2
form
N
(3.1) A− F AF∗ = GJG∗,
k k
X
k=1
where F ∈ L(G), k = 1,... ,N, are given contractions on the Hilbert space G. Also
k
G = U V ∈ L(E ⊕ E ,G) and J = I ⊕ I . The wave operators associated
1 2 E1 E2
to (3.1(cid:2)) are in(cid:3)troduced by the formulae: U∗ = [U ]∞ , V∗ = [V ]∞ , where U =
∞ k k=0 ∞ k k=0 k
[F U] : H ⊗E → G, V = [F V] : H ⊗E → G. We will assume that both
σ |σ|=k k 1 k σ |σ|=k k 2
8
U and V are bounded and also that lim kF∗gk = 0 for all g ∈ G. Under
∞ ∞ k→∞ |σ|=k σ
these assumptions we deduce that (3.1) has a uPnique solution given by
(3.2) A = U∗ U −V∗V .
∞ ∞ ∞ ∞
The next result was already noticed in [5], but here we present a different approach
based on system theoretic ideas.
Theorem 3.1. The solution (3.2) of the displacement equation (3.1) is positive if and
only if there exists T ∈ S(H,E ,E ) such that V = TU .
1 2 ∞ ∞
Proof. Assume A = U∗ U −V∗V ≥ 0 and let A = LL∗ be a factorization of A with
∞ ∞ ∞ ∞
L ∈ L(F,G) for some Hilbert space F. From (3.1) we deduce that
N
LL∗ +VV∗ = F LL∗F∗ +UU∗.
k k
X
k=1
In matrix form,
L∗F∗
1
.
L∗  .. 
(3.3) L V = F L ... F L U .
(cid:20) V∗ (cid:21) 1 N L∗F∗
(cid:2) (cid:3) (cid:2) (cid:3) N 
 U∗ 
 
Define A∗ = L V and B∗ = F L ... F L U , then we deduce from (3.3)
1 N
thatthereexis(cid:2)ts anun(cid:3)itaryoperato(cid:2)rθ ∈ L(R(B),R(A))(cid:3)such thatA = θ B.Itfollows
0 0
that there exist Hilbert spaces R , R , and an unitary extension θ ∈ L(F⊕N ⊕E ⊕
1 2 1
R ,F ⊕E ⊕R ) of θ , hence this extension satisfies the relation
1 2 2 0
A B
(3.4) = θ .
(cid:20) 0 (cid:21) (cid:20) 0 (cid:21)
R2 R1
Let θ , i ∈ {1,2,3}, j ∈ {1,2,... ,N + 2}, be the matrix coefficients of θ. It is
ij
convenient to rename some of these coefficients. Thus, we set
X = θ , k = 1,...N, Z = θ ,
k 1k 1,N+1
Y = θ , k = 1,...N, W = θ .
k 2k 2,N+1
From (3.4) we deduce that
N
L∗ = X L∗F∗ +ZU∗
k k
X
k=1
and
N
V∗ = Y L∗F∗ +WV∗.
k k
X
k=1
9
By induction we deduce that
N n
(3.5) V∗ = WU∗ + Y X ZU∗F∗ + Q L∗F∗,
k σ kσ τ τ
X X X
k=1|σ|=0 |τ|=n+1
where Q are monomials of length |τ| in the variables X , ...,X , Y , ..., Y . Since
τ 1 N 1 N
θ is unitary it follows that all Q are contractions.
τ
We define T = W and for j > 0,
00
T = [Y X Z] .
0j k σ |σ|=j−1;k=1,...,N
Then we define T , i > 0, j ≥ i, by the formula (2.1) and T = 0 for i > j. We show
ij ij
that T = [T ]∞ belongs to S(H,E ,E ). To that end, we introduce the notation: for
ij i,j=0 1 2
n ≥ 0,
⊕Nn ⊕Nn
X (n) = X ... X , Y (n) = Y ... Y ,
1 1 N 1 1 N
(cid:2) (cid:3) (cid:2) (cid:3)
X (n) = Z⊕Nn, Y (n) = W⊕Nn.
2 2
We check by induction that T = Y (0), T = Y (0)X (1), and for j ≥ 2,
00 2 01 1 2
T = Y (0)X (1)...X (j −1)X (j).
0j 1 1 1 2
Consequently, T∗ is the transfer map of the linear time variant system
x(n+1) = X (n)∗x(n)+Y (n)∗u(n)
1 1

 y(n) = X (n)∗x(n)+Y (n)∗u(n), n ≥ 0.
2 2
 X (n) X (n)
Sinceeachmatrix 1 2 , n ≥ 0,isacontraction, itfollowsthatT∗ isacon-
(cid:20) Y (n) Y (n) (cid:21)
1 2
traction,henceT isacontractionandT ∈ S(H,E ,E ). Also,sincelim kF∗gk =
1 2 k→∞ |σ|=k σ
0 for all g ∈ G, we deduce from (3.5) that V = TU . P
∞ ∞
This result is a generalization of Theorem 3.1. It can be used to solve interpolation
and moment problems in several non commuting variables as suggested in [5]. The
proofgivenhereemphasizes thefactthatthistypeofproblemsappearsasaninteresting
particular case of similar questions in time variant setting (as described, for instance,
in [4]). Next we develop this idea by giving a parametrization of the elements in
S(H,E ,E ).
1 2
3.1. Schur Parameters. We introduce a Schur type parametrization for S(H,E ,E )
1 2
and then give a number of applications. A first instance of such a result might be con-
sidered Euler’s description ofSO(3). Schur’s classical result in[13]gives a parametriza-
tion of the contractive holomorphic functions on the unit disk and Szeg¨o’s theory of
orthogonal polynomials provides an alternative to this result for probability measures
on the unit circle.
Here we provide a refinement of a parametrization of S(H,E ,E ) given in [11]. We
1 2
can deal with a slightly more general situation than S(H). For a contraction C we
10