Characteristic Functions for Ergodic Tuples
Santanu Dey and Rolf Gohm
Abstract. Motivated by a result on weak Markov dilations, we define a notion
of characteristic function for ergodic and coisometric row contractions with a
one-dimensional invariant subspace for the adjoints. This extends a definition
given by G.Popescu. We prove that our characteristic function is a complete
unitary invariant for such tuples and show how it can be computed.
Mathematics Subject Classification (2000).
Primary 47A20, 47A13; Secondary 46L53, 46L05.
Keywords. completely positive, dilation, conjugacy, ergodic, coisometric, row
contraction, characteristic function, Cuntz algebra.
0. Introduction
If Z = summationtextdi=1 Ai ? A?i is a normal, unital, ergodic, completely positive map on
B(H), the bounded linear operators on a complex separable Hilbert space, and if
there is a (necessarily unique) invariant vector state for Z, then we also say that
A = (A1,...,Ad) is a coisometric, ergodic row contraction with a one-dimensional
invariant subspace for the adjoints. Precise definitions are given below. This is the
main setting to be investigated in this paper.
In Section 1 we give a concise review of a result on the dilations of Z obtained
by R.Gohm in [Go04] in a chapter called ?Cocycles and Coboundaries?. There
exists a conjugacy between a homomorphic dilation of Z and a tensor shift, and
we emphasize an explicit infinite product formula that can be obtained for the
intertwining unitary. [Go04] may also be consulted for connections of this topic
to a scattering theory for noncommutative Markov chains by B.K?ummerer and
H.Maassen (cf. [KM00]) and more general for the relevance of this setting in
applications.
In this work we are concerned with its relevance in operator theory and
correspondingly in Section 2 we shift our attention to the row contraction A =
(A1,...,Ad). Our starting point has been the observation that the intertwining
unitary mentioned above has many similarities with the notion of characteristic
2 Santanu Dey and Rolf Gohm
function occurring in the theory of functional models of contractions, as initiated
by B.Sz.-Nagy and C.Foias (cf. [NF70, FF90]). In fact, the center of our work
is the commuting diagram 3.3 in Section 3, which connects the results in [Go04]
mentioned above with the theory of minimal isometric dilations of row contractions
by G.Popescu (cf. [Po89a]) and shows that the intertwining unitary determines
a multi-analytic inner function, in the sense introduced by G.Popescu in [Po89c,
Po95]. We call this inner function the extended characteristic function of the tuple
A, see Definition 3.3.
Section 4 is concerned with an explicit computation of this inner function.
In Section 5 we show that it is an extension of the characteristic function of the
?-stable part ?A of A, the latter in the sense of Popescu?s generalization of the
Sz.-Nagy-Foias theory to row contractions (cf. [Po89b]). This explains why we call
our inner function an extended characteristic function. The row contraction A is a
one-dimensional extension of the ?-stable row contraction ?A, and in our analysis
we separate the new part of the characteristic function from the part already given
by Popescu.
G.Popescu has shown in [Po89b] that for completely non-coisometric tuples,
in particular for ?-stable ones, his characteristic function is a complete invariant
for unitary equivalence. In Section 6 we prove that our extended characteristic
function does the same for the tuples A described above. In this sense it is char-
acteristic. This is remarkable because the strength of Popescu?s definition lies in
the completely non-coisometric situation while we always deal with a coisometric
tuple A. The extended characteristic function also does not depend on the choice
of the decompositionsummationtextdi=1 Ai?A?i of the completely positive map Z and hence also
characterizes Z up to conjugacy. We think that together with its nice properties
established earlier this clearly indicates that the extended characteristic function
is a valuable tool for classifying and investigating such tuples respectively such
completely positive maps.
Section 7 contains a worked example for the constructions in this paper.
1. Weak Markov dilations and conjugacy
In this section we give a brief and condensed review of results in [Go04], Chapter
2, which will be used in the following and which, as described in the introduction,
motivated the investigations documented in this paper. We also introduce notation.
A theory of weak Markov dilations has been developed in [BP94]. For a (sin-
gle) normal unital completely positive map Z : B(H) ? B(H), where B(H)
consists of the bounded linear operators on a (complex, separable) Hilbert space,
it asks for a normal unital ??endomorphism ?J : B( ?H) ? B( ?H), where ?H is a
Hilbert space containing H, such that for all n ?N and all x ? B(H)
Zn(x) = pH ?Jn(xpH) |H.
Characteristic Functions for Ergodic Tuples 3
Here pH is the orthogonal projection onto H. There are many ways to construct
?J. In [Go04], 2.3, we gave a construction analogous to the idea of ?coupling to a
shift? used in [K?u85] for describing quantum Markov processes. This gives rise to
a number of interesting problems which remain hidden in other constructions.
We proceed in two steps. First note that there is a Kraus decomposition
Z(x) = summationtextdi=1 ai xa?i with (ai)di=1 ? B(H). Here d = ? is allowed in which case
the sum should be interpreted as a limit in the strong operator topology. Let P
be a d-dimensional Hilbert space with orthonormal basis {epsilon11,...,epsilon1d}, further K
another Hilbert space with a distinguished unit vector ?K ? K. We identify H
with H??K ? H?K and again denote by pH the orthogonal projection onto H.
For K large enough there exists an isometry
u : H?P ? H?K s.t. pHu(h?epsilon1i) = ai(h),
for all h ? H, i = 1,...,d, or equivalently,
u?(h??K) =
dsummationdisplay
i=1
a?i(h)?epsilon1i.
Explicitly, one may take K = Cd+1 (resp. infinite-dimensional) and identify
H?K similarequal (H??K)?
dcircleplusdisplay
1
H similarequal H?
dcircleplusdisplay
1
H.
Then, using isometries u1,...,ud : H ? H?circleplustextd1 H with orthogonal ranges
and such that ai = pHui for all i (for example, such isometries are explicitly
constructed in Popescu?s formula for isometric dilations, cf. [Po89a] or equation
3.2 in Section 3), we can define
u(h?epsilon1i) := ui(h)
for all h ? H, i = 1,...,d and check that u has the desired properties. Now we
define a ??homomorphism
J : B(H) ? B(H?K),
x mapsto? u(x?1P)u?.
It satisfies
pHJ(x)(h??K) = pHu(x?1)u?(h??K)
= pHu(x?1)parenleftbig
dsummationdisplay
i=1
a?i(h)?epsilon1iparenrightbig=
dsummationdisplay
i=1
ai xa?i(h) = Z(x)(h),
which means that J is a kind of first order dilation for Z.
4 Santanu Dey and Rolf Gohm
For the second step we write ?K := circlemultiplytext?1 K for an infinite tensor product of
Hilbert spaces along the sequence (?K) of unit vectors in the copies of K. We have
a distinguished unit vector ??K and a (kind of) tensor shift
R : B(?K) ? B(P ? ?K), ?y mapsto?1P ? ?y.
Finally ?H := H? ?K and we define a normal ??endomorphism
?J : B( ?H) ? B( ?H),
B(H)?B(?K) owner x? ?y mapsto? J(x)? ?y ? B(H?K)?B(?K).
Here we used von Neumann tensor products and (on the right hand side) a shift
identification K? ?K similarequal ?K. We can also write ?J in the form
?J(?) = u(IdH ?R)(?)u?,
where u is identified with u?1?K. The natural embedding H similarequal H???K ? ?H leads
to the restriction ?J := ?J| ?H with ?H := spann?0 ?Jn(pH)( ?H), which can be checked to
be a normal unital ?-endomorphism satisfying all the properties of a weak Markov
dilation for Z described above. See [Go04], 2.3.
A Kraus decomposition of ?J can be written as
?J(x) =
dsummationdisplay
i=1
ti xt?i,
where ti ? B( ?H) is obtained by linear extension of H? ?K owner h? ?k mapsto? ui(h)? ?k =
u(h?epsilon1i)??k ? (H?K)??K similarequal H??K. Because ?J is a normal unital ??endomorphism
the (ti)di=1 generate a representation of the Cuntz algebra Od on ?H which we called
a coupling representation in [Go04], 2.4. Note that the tuple (t1,...,td) is an iso-
metric dilation of the tuple (a1,...,ad), i.e., the ti are isometries with orthogonal
ranges and pHtni |H = ani for all i = 1,...,d and n ?N.
The following multi-index notation will be used frequently in this work.
Let ? denote the set {1,2,...,d}. For operator tuples (a1,...,ad), given ? =
(?1,...,?m) in ?m, a? will stand for the operator a?1a?2 ...a?m, |?| := m.
Further ?? := ??n=0?n, where ?0 := {0} and a0 is the identity operator. If we write
a?? this always means (a?)? = a??m ...a??1.
Back to our isometric dilation, it can be checked that
span{t?h : h ? H,? ? ??} = ?H,
which means that we have a minimal isometric dilation, cf. [Po89a] or the begin-
ning of Section 3. For more details on the construction above see [Go04], 2.3 and
2.4.
Assume now that there is an invariant vector state for Z : B(H) ? B(H)
given by a unit vector ?H ? H. Equivalent: There is a unit vector ?P =summationtextdi=1 ?iepsilon1i ?
P such that u(?H ? ?P) = ?H ? ?K. Also equivalent: For i = 1,...,d we have
Characteristic Functions for Ergodic Tuples 5
a?i ?H = ?i ?H. Here ?i ?Cwithsummationtextdi=1|?i|2 = 1 and we used complex conjugation
to get nice formulas later. See [Go04], A.5.1, for a proof of the equivalences.
On ?P := circlemultiplytext?1 P along the unit vectors (?P) in the copies of P we have a
tensor shift
S : B( ?P) ? B( ?P), ?y mapsto?1P ? ?y.
Its Kraus decomposition is S(?y) =summationtextdi=1 si ?ys?i with si ? B( ?P) and si(?k) = epsilon1i ??k
for ?k ? ?P and i = 1,...,d. In [Go04], 2.5, we obtained an interesting description
of the situation when the dilation ?J is conjugate to the shift endomorphism S.
This result will be further analyzed in this paper. We give a version suitable for
our present needs but the reader should have no problems to obtain a proof of the
following from [Go04], 2.5.
Theorem 1.1. Let Z : B(H) ? B(H) be a normal unital completely positive map
with an invariant vector state ??H,??H?. Notation as introduced above, d ? 2.
The following assertions are equivalent:
(a) Z is ergodic, i.e., the fixed point space of Z consists of multiples of the iden-
tity.
(b) The vector state ??H,??H? is absorbing for Z, i.e., if n ? ? then ?(Zn(x)) ?
??H,x?H? for all normal states ? and all x ? B(H). (In particular, the in-
variant vector state is unique.)
(c) ?J and S are conjugate, i.e., there exists a unitary w : ?H ? ?P such that
?J(?x) = w?S(w?xw?)w.
(d) The Od?representations corresponding to ?J and S are unitarily equivalent,
i.e.,
wti = siw for i = 1,...,d.
An explicit formula can be given for an intertwining unitary as occurring in (c) and
(d). If any of the assertions above is valid then the following limit exists strongly,
?w = limn??u?0n ...u?01 : H? ?K ? H? ?P,
where we used a leg notation, i.e., u0n = (IdH ?R)n?1(u). In other words u0n is
u acting on H and on the n?th copy of P. Further ?w is a partial isometry with
initial space ?H and final space ?P similarequal ?H ? ?P ? H? ?P and we can define w as the
corresponding restriction of ?w.
To illustrate the product formula for w, which will be our main interest in
this work, we use it to derive (d).
wti(h? ?k) = wbracketleftbigu(h?epsilon1i)? ?kbracketrightbig= limn??u?0n ...u?01u01(h?epsilon1i ? ?k)
= limn??u?0n ...u?02(h?epsilon1i ? ?k) = siw(h? ?k).
Let us finally note that Theorem 1.1 is related to the conjugacy results in [Pow88]
and [BJP96]. Compare also Proposition 2.4.
6 Santanu Dey and Rolf Gohm
2. Ergodic coisometric row contractions
In the previous section we considered a map Z : B(H) ? B(H) given by Z(x) =summationtext
d
i=1 Ai xA
?i, where (Ai)di=1 ? B(H). We can think of (Ai)di=1 as a d-tuple A =
(A1,...,Ad) or (with the same notation) as a linear map
A = (A1,...,Ad) :
dcircleplusdisplay
i=1
H ? H.
(Concentrating now on the tuple we have changed to capital letters A. We will
sometimes return to lower case letters a when we want to emphasize that we are
in the (tensor product) setting of Section 1.) We have the following dictionary.
Z(1) ?1 ?
dsummationdisplay
i=1
Ai A?i ?1
? A is a contraction
Z(1) = 1 ?
dsummationdisplay
i=1
Ai A?i = 1
parenleftbigZ is called unitalparenrightbig parenleftbigA is called coisometricparenrightbig
??H,??H? = ??H,Z(?)?H? ? A?i ?H = ?i ?H, ?i ?C,
dsummationdisplay
i=1
|?i|2 = 1
parenleftbig invariant vector stateparenrightbig parenleftbig common eigenvector for adjointsparenrightbig
Z ergodic ? {Ai,A?i}prime = C1parenleftbig
trivial fixed point spaceparenrightbig parenleftbigtrivial commutantparenrightbig
The converse of the implication at the end of the dictionary is not valid. This
is related to the fact that the fixed point space of a completely positive map is
not always an algebra. Compare the detailed discussion of this phenomenon in
[BJKW00].
By a slight abuse of language we call the tuple (or row contraction) A =
(A1,...,Ad) ergodic if the corresponding map Z is ergodic. With this terminology
we can interpret Theorem 1.1 as a result about ergodic coisometric row contrac-
tions A with a common eigenvector ?H for the adjoints A?i. This will be examined
starting with Section 3. To represent these objects more explicitly let us write
?H:= HcircleminusC?
H. With respect to the decomposition H = C?H?
?H we get 2?2?
block matrices
Ai =
parenleftbigg ?
i 0
|lscripti? ?Ai
parenrightbigg
, A?i =
parenleftbigg ?
i ?lscripti|
0 ?A?i
parenrightbigg
. (2.1)
Characteristic Functions for Ergodic Tuples 7
Here ?Ai ? B( ?H) and lscripti ??H. For the off-diagonal terms we used a Dirac
notation that should be clear without further comments.
Note that the case d = 1 is rather uninteresting in this setting because if A
is a coisometry with block matrix
parenleftbigg ? 0
|lscript? ?A
parenrightbigg
then because
parenleftbigg 1 0
0 1
parenrightbigg
= AA? =
parenleftbigg |?|2 ??lscript|
?|lscript? |lscript??lscript|+ ?A?A?
parenrightbigg
we always have lscript = 0. But for d ? 2 there are many interesting examples arising
from unital ergodic completely positive maps with invariant vector states. See
Section 1 and also Section 7 for an explicit example. We always assume d ? 2.
Proposition 2.1. A coisometric row contraction A = (A1,...,Ad) is ergodic with
common eigenvector ?H for the adjoints A?1,...,A?d if and only if ?H is invariant
for A1,...,Ad and the restricted row contraction (?A1,...,?Ad) on ?H is ?-stable,
i.e., for all h ??H
limn??
summationdisplay
|?|=n
bardbl?A??hbardbl2 = 0 .
Here we used the multi-index notation introduced in Section 1. Note that
?-stable tuples are also called pure, we prefer the terminology from [FF90].
Proof. It is clear that ?H is a common eigenvector for the adjoints if and only if ?H
is invariant for A1,...,Ad. Let Z(?) = summationtextdi=1 Ai ?A?i be the associated completely
positive map. With q := 1?|?H???H|, the orthogonal projection onto ?H, and by
using qAi q = Ai q similarequal ?Ai for all i, we get
Zn(q) =
summationdisplay
|?|=n
A? qA?? =
summationdisplay
|?|=n
?A? ?A??
and thus for all h ??H summationdisplay
|?|=n
bardbl?A??hbardbl2 = ?h,Zn(q)h?.
Now it is well known that ergodicity of Z is equivalent to Zn(q) ? 0 for n ? ?
in the weak operator topology. See [GKL06], Prop. 3.2. This completes the proof.
a50
Remark 2.2. Given a coisometric row contraction a = (a1,...,ad) we also have
the isometry u : H?P ? H?K from Section 1. We introduce the linear map
a : P ? B(H), k mapsto? ak defined by
a?k(h)?k := (1H ?|k??k|)u?(h??K).
Compare [Go04], A.3.3. In particular ai = aepsilon1i for i = 1,...,d, where {epsilon11,...,epsilon1d} is
the orthonormal basis of P used in the definition of u. Arveson?s metric operator
8 Santanu Dey and Rolf Gohm
spaces, cf. [Ar03], give a conceptual foundation for basis transformations in the
operator space linearly spanned by the ai. Similarly, in our formalism a unitary in
B(P) transforms a = (a1,...,ad) into another tuple aprime = (aprime1,...,aprimed). If ?H is a
common eigenvector for the adjoints a?i then ?H is also a common eigenvector for
the adjoints (aprimei)? but of course the eigenvalues are transformed to another tuple
?prime = (?prime1,...,?primed). We should consider the tuples a and aprime to be essentially the
same. This also means that the complex numbers ?i are not particularly important
and they should not play a role in classification. They just reflect a certain choice
of orthonormal basis in the relevant metric operator space. Independent of basis
transformations is the vector ?P = summationtextdi=1 ?i epsilon1i ? P satisfying u(?H ? ?P) =
?H ??K (see Section 1) and the operator a?P =summationtextdi=1 ?i ai.
For later use we show
Proposition 2.3. Let A = (A1,...,Ad) be an ergodic coisometric row contraction
such that A?i ?H = ?i ?H for all i, further A?P := summationtextdi=1 ?i Ai. Then for n ? ?
in the strong operator topology
(A??P)n ? |?H???H|.
Proof. We use the setting of Section 1 to be able to apply Theorem 1.1. From
u?(h??K) =summationtextdi=1 a?i(h)?epsilon1i we obtain
u?(h??K) = a??P(h)??P ?hprime
with hprime ? H ? ??P. Assume that h ??H. Because u? is isometric on H ? ?K we
conclude that
u?(?H ??K) = ?H ??P ? u?(h??K) (2.2)
and thus also a??P(h) ??H. In other words,
a??P( ?H) ? ?H .
Let qn be the orthogonal projection from H ?circlemultiplytextn1 P onto ?H ?circlemultiplytextn1 P. From
Theorem 1.1 it follows that
(1?qn)u?0n ...u?01(h?
ncirclemultiplydisplay
1
?K) ? 0 (n ? ?).
On the other hand, by iterating the formula from the beginning,
u?0n ...u?01(h?
ncirclemultiplydisplay
1
?K) =parenleftbig(a??P)n(h)?
ncirclemultiplydisplay
1
?Pparenrightbig?hprime
with hprime ? H?(circlemultiplytextn1 ?P)?. It follows that also
(1?qn)parenleftbig(a??P)n(h)?
ncirclemultiplydisplay
1
?Pparenrightbig? 0.
Characteristic Functions for Ergodic Tuples 9
But from a??P( ?H) ? ?H we have qnparenleftbig(a??P)n(h)?circlemultiplytextn1 ?Pparenrightbig= 0 for all n. We conclude
that (a??P)n(h) ? 0 for n ? ?. Further
a??P?H =
dsummationdisplay
i=1
?i a?i ?H =
dsummationdisplay
i=1
?i ?i ?H = ?H,
and the proposition is proved. a50
The following proposition summarizes some well known properties of minimal
isometric dilations and associated Cuntz algebra representations.
Proposition 2.4. Suppose A is a coisometric tuple on H and V is its minimal
isometric dilation. Assume ?H is a distinguished unit vector in H and ? =
(?1,...,?d) ?Cd, summationtexti|?i|2 = 1. Then the following are equivalent.
1. A is ergodic and A?i ?H = ?i ?H for all i.
2. V is ergodic and V?i ?H = ?i ?H for all i.
3. V?i ?H = ?i ?H and V generates the GNS-representation of the Cuntz algebra
Od = C?{g1,??? ,gd} (gi its abstract generators) with respect to the Cuntz
state which maps
g? g?? mapsto? ?? ??, ??,? ? ??.
Cuntz states are pure and the corresponding GNS-representations are irreducible.
This Proposition clearly follows from Theorem 5.1 of [BJKW00], Theorem
3.3 and Theorem 4.1 of [BJP96]. Note that in Theorem 1.1(d) we already saw a
concrete version of the corresponding Cuntz algebra representation.
3. A new characteristic function
First we recall some more details of the theory of minimal isometric dilations for
row contractions (cf. [Po89a]) and introduce further notation.
The full Fock space over Cd (d ? 2) denoted by ?(Cd) is
?(Cd) := C?Cd ?(Cd)?2 ?????(Cd)?m ???? .
1?0???? is called the vacuum vector. Let{e1,...,ed}be the standard orthonormal
basis of Cd. Recall that we include d = ? in which case Cd stands for a complex
separable Hilbert space of infinite dimension. For ? ? ??, e? will denote the vector
e?1 ?e?2 ?????e?m in the full Fock space ?(Cd) and e0 will denote the vacuum
vector. Then the (left) creation operators Li on ?(Cd) are defined by
Lix = ei ?x
for 1 ? i ? d and x ? ?(Cd). The row contraction L = (L1,...,Ld) consists of
isometries with orthogonal ranges.
10 Santanu Dey and Rolf Gohm
Let T = (T1,??? ,Td) be a row contraction on a Hilbert space H. Treating
T as a row operator from circleplustextdi=1H to H, define D? := (1?TT?)12 : H ? H and
D := (1?T?T)12 :circleplustextdi=1H ?circleplustextdi=1H. This implies that
D? = (1?
dsummationdisplay
i=1
TiT?i )12, D = (?ij1?T?i Tj)12d?d. (3.1)
Observe that TD2 = D2?T and hence TD = D?T. Let D := Range D and
D? := Range D?. Popescu in [Po89a] gave the following explicit presentation of
the minimal isometric dilation of T by V on H?(?(Cd)?D),
Vi(h?
summationdisplay
????
e? ?d?) = Tih?[e0 ?Dih+ei ?
summationdisplay
????
e? ?d?] (3.2)
for h ? H and d? ? D. Here Dih := D(0,...,0,h,0,...,0) and h is embedded at
the ith component.
In other words, the Vi are isometries with orthogonal ranges such that T?i =
V?i |H for i = 1,...,d and the spaces V?H with ? ? ?? together span the Hilbert
space on which the Vi are defined. It is an important fact, which we shall use
repeatedly, that such minimal isometric dilations are unique up to unitary equiv-
alence (cf. [Po89a]).
Now, as in Section 2, let A = (A1,??? ,Ad), Ai ? B(H), be an ergodic
coisometric tuple with A?i?H = ?i?H for some unit vector ?H ? H and some
? ? Cd, summationtexti|?i|2 = 1. Let V = (V1,??? ,Vd) be the minimal isometric dilation of
A given by Popescu?s construction (see equation 3.2) onH?parenleftbig?(Cd)?DAparenrightbig. Because
A?i = V?i |H we also have V?i ?H = ?i?H and because V generates an irreducible
Od?representation (Proposition 2.4), we see that V is also a minimal isometric
dilation of ? : Cd ?C. In fact, we can think of ? as the most elementary example
of a tuple with all the properties stated for A. Let ?V = (?V1,??? , ?Vd) be the minimal
isometric dilation of ? given by Popescu?s construction on C?(?(Cd)?D?).
Because A is coisometric it follows from equation 3.1 that D is in fact a
projection and hence D = (?ij1?A?iAj)d?d. We infer that D(A?1,??? ,A?d)T = 0,
where T stands for transpose. Applied to ? instead of A this shows that D? =
(1?|????|) and
D? ?C(?1,??? ,?d)T = Cd,
where ? = (?1,??? ,?d).
Remark 3.1. Because ?H is cyclic for {V?, ? ? ??} we have
span{A??H : ? ? ??} = span{pHV??H : ? ? ??} = H.
Using the notation from equation 2.1 this further implies that
span{?A? li : ? ? ??,1 ? i ? d} = ?H .
Characteristic Functions for Ergodic Tuples 11
As minimal isometric dilations of the tuple ? are unique up to unitary equi-
valence, there exists a unitary
W : H?(?(Cd)?DA) ?C?(?(Cd)?D?),
such that WVi = ?ViW for all i.
After showing the existence of W we now proceed to compute W explicitly.
For A, by using Popescu?s construction, we have its minimal isometric dilation V
on H?(?(Cd)?DA). Another way of constructing a minimal isometric dilation t
of a was demonstrated in Section 1 on the space ?H (obtained by restricting to the
minimal subspace of H? ?K with respect to t). Identifying A and a on the Hilbert
space H there is a unitary ?A : ?H ? H?(?(Cd)?DA) which is the identity on
H and satisfies Vi?A = ?Ati.
By Theorem 1.1(d) the tuple s on ?P arising from the tensor shift is unitarily
equivalent to t (resp. V), explicitly wti = siw for all i. An alternative view-
point on the existence of w is to note that s is a minimal isometric dilation of
?. In fact, s?i ??P = ?epsilon1i,?P???P = ?i ??P for all i. Hence there is also a unitary
?? : ?P ?C?(?(Cd)?D?) with ????P = 1 ?C which satisfies ?Vi?? = ??si.
Remark 3.2. It is possible to describe ?? in an explicit way and in doing so
to construct an interesting and natural (unitary) identification of circlemultiplytext?1 Cd and
C?(?(Cd)?Cd?1). In fact, recall (from Section 1) that ?P =circlemultiplytext?1 P and the space
P is nothing but a d-dimensional Hilbert space. Hence we can identify
Cd similarequal P = ?P ?C?P similarequal D? ?C?T similarequalCd?1 ?C
In this identification the orthonormal basis (epsilon1i)di=1 of P goes to the canonical basis
(ei)di=1 of Cd, in particular the vector ?P = summationtexti ?i epsilon1i goes to ?T = (?1,??? ,?d)T
and we have ?Psimilarequal D?. Then we can write
?? : ??P mapsto? 1 ?C,
k ???P mapsto? e0 ?k
epsilon1? ?k ???P mapsto? e? ?k,
where k ??P, ? ? ??, epsilon1? = epsilon1?1?...epsilon1?n ?circlemultiplytextn1 P (the first n copies ofP in the infinite
tensor product ?P), e? = e?1 ?...e?n ? ?(Cd) as usual. It is easily checked that
?? given in this way indeed satisfies the equation ?Vi?? = ??si (for all i), which
may thus be seen as the abstract characterization of this unitary map (together
with ????P = 1).
Summarizing, for i = 1,...,d
Vi ?A = ?A ti, wti = siw, ?Vi ?? = ?? si
12 Santanu Dey and Rolf Gohm
and we have the commuting diagram
?H w d47d47
?A
d15d15
?P
??
d15d15
H?(?(Cd)?DA) W d47d47 C?(?(Cd)?D?).
(3.3)
From the diagram we get
W = ??w??1A .
Combined with the equations above this yields WVi = ?Vi W and we see that W
is nothing but the dilations-intertwining map which we have already introduced
earlier. Hence w and W are essentially the same thing and for the study of certain
problems it may be helpful to switch from one picture to the other.
In the following we analyze W to arrive at an interpretation as a new kind
of characteristic function. First we have an isometric embedding
?C := W|H : H ?C?(?(Cd)?D?). (3.4)
Note that ?C ?H = W ?H = 1 ?C. The remaining part is an isometry
M?? := W|?(Cd)?DA : ?(Cd)?DA ? ?(Cd)?D?. (3.5)
From equation 3.2 we get for all i
Vi|?(Cd)?DA = (Li ?1DA),
?Vi|?(Cd)?D
? = (Li ?1D?),
and we conclude that
M??(Li ?1DA) = (Li ?1D?)M??, ?1 ? i ? d. (3.6)
In other words, M?? is a multi-analytic inner function in the sense of [Po89c, Po95].
It is determined by its symbol
?? := W|e
0?DA : DA ? ?(C
d)?D?, (3.7)
where we have identified e0?DA and DA. In other words, we think of the symbol
?? as an isometric embedding of DA into ?(Cd)?D?.
Definition 3.3. We call M?? (or ??) the extended characteristic function of the row
contraction A,
See Sections 5 and 6 for more explanation and justification of this terminol-
ogy.
Characteristic Functions for Ergodic Tuples 13
4. Explicit computation of the extended characteristic function
To express the extended characteristic function more explicitly in terms of the
tuple A we start by defining
?D? : ?H= HcircleminusC?H ? ?P = P circleminusC?P similarequal D?, (4.1)
h mapsto?parenleftbig??H|?1Pparenrightbigu?(h??K),
where u : H?P ? H?K is the isometry introduced in Section 1. That indeed the
range of ?D? is contained in ?P follows from equation 2.2, i.e., u?(h??K) ? ?H??P
for h ??H. With notations from equation 2.1 we can get a more concrete formula.
Lemma 4.1. For all h ??H we have ?D?(h) =summationtextdi=1?lscripti,h?epsilon1i.
Proof. parenleftbig??H|?1Pparenrightbigu?(h??K) =summationtextdi=1??H,a?ih??epsilon1i =summationtextdi=1?lscripti,h?epsilon1i. a50
Proposition 4.2. The map ?C : H ? C?(?(Cd)?D?) from equation 3.4 is given
explicitly by ?C?H = 1 and for h ??H by
?Ch = summationdisplay
????
e? ? ?D??A??h.
Proof. As W?H = 1 also ?C?H = 1. Assume h ??H. Then
u01(h???K) =
summationdisplay
i
a?ih?epsilon1i ???K
=
summationdisplay
i
?lscripti,h??H ?epsilon1i ???K +
summationdisplay
i
?a?
i h?epsilon1i ???K.
Because u?(?H ??K) = ?H ??P we obtain (with Lemma 4.1) for the first part
limn??u?0n???u?02(
summationdisplay
i
?lscripti,h??H ?epsilon1i ???K)
=
summationdisplay
i
?lscripti,h??H ?epsilon1i ???P = ?H ? ?D?h???P similarequal ?D?h???P ? ?P.
Using the product formula from Theorem 1.1 and iterating the argument above
we get
?C(h) = Wh = ??w??1A (h)
= ??( ?D?h???P) + ?? limn??u?0n???u?02
summationdisplay
i
?a?
i h?epsilon1i ???K
= e0 ? ?D?h + ?? limn??u?0n???u?03
summationdisplay
j,i
parenleftbig?lscript
j,
?a?
i h??H+
?a?
j
?a?
i h
parenrightbig?epsilon1
i ?epsilon1j ???K
= e0 ? ?D?h+
dsummationdisplay
i=1
ei ? ?D? ?a?i h + ?? limn??u?0n???u?03
summationdisplay
j,i
?a?
j
?a?
i h?epsilon1i ?epsilon1j ?? ?K
14 Santanu Dey and Rolf Gohm
= ...
=
summationdisplay
|?|