Download The Characteristic Functions of Operator

Functionsof Operator
The Characteristic
Theoryand ElectricalNetwork Realization
J. WILLIAM HELTON
one of the main results in Nagy and Foias [N-F] says that any analytic
function 0(z) which is defined on the unit disk and whose values are operators
of norm less than or equal to one on a Hilbert spacecoincideswith the characteristic function of some operator ? on another Hilbert spaceH (Theorem
vI, 3.1 tN-Fl). Amazingly enough a very specialcaseof this abstract theorem
was developedindependently by engineersand is one formulation of the main
realizability theorem in electrical engineering.The purpose of this note is to
sketch very roughly how this occurs.
The fact that a connection exists between modern operator theory and
electrical engineeringhas been known for severalyears. In his book [L] Livsic
worked out the connection between his approach to characteristic functions
for operators and network realizability theory. Unfortunately, this book has
never been translated into English and it has circulated little among American
operator theorists. Also, DeWilde in [D] relates mathematics developed by
Livsic and Potopov to network synthesis.There has been work by Sandburg,
Saeks and Levan in a very different direction involving the original Halmos
dilation for a contraction operator and the synthesis of a lossy network with
a losslessnetwork and a resistor.
This paper takes the Lax-Phillips [L-P] and Nagy-Foias viewpoint to
operator theory rather than that of Livsic and the interpretation given here
to the Nagy-Foias representationtheorem appearsto be new.
The first section is an exposition intended for operator theorists of some
basic electric network theory, in the second part we relate this to operator
theory. For a desoiption of electric network theory see[W] or [N].
The author wishesto thank A. H. Zemanianfor his encouragementand for
discussionswhich were most helpful.
A brief expositionof electrical network theory. An electric network (n port)
is thought of as a box with rupairs of terminals on it. For simplicity sake consider a network gl which is a 1 port. If one connectsa voltage sourcewith voltage
0 acrossthe terminals of fl, then a unique current i flows through the circuit
(seeFigure 1). Yoltage and current are
403
Indiana
University
Mathematics Journal, Vol. 22, No. 5 (1972)
4p.4
J. W. IIELTON
t port
Frcunn1
both functiong a(l) and i(t) of time and are assumedto be complex valued
functions in L'[- -, * - ]. The set fi, of all pairs [u(r),f(r)] possiblefor a network
9l uniquely determines9r. w-e shall call fi the graph of the function eL
By comparing the network s(, to the network consistingof a singreresistor
of resistancer electrical engineersarrive at an interpretation of.$[r-'7'u + r""d
and jlr-'/'u - ,"'if as incoming and outgoing signalsrespectively.They usually
take the referenceresistancer to be I and define a "scattering operator,, s for
the network st,by
S*(u*,):+(u-il.
Every network which we consider here will have a 'lscattering operator" I
which is a boundedlinear map of.L'[- -, f *] into -L'[- -, * -]. The study
of scattering operators for networks is equivalent to the study of networks,
at least from a "black box" point of view.
All networks which we consider will be time invariant and they will have
the property that no signal comesout before time counless a signal has been
put in before time 16. The secondproperty is called causality.Mathematically,
time invariancemeansthat S commuteswith translation by time, and causality
meansthat S maps the subspacef,'fto , -] of -L'[- -, -] into itself. If one
Fourier transform functions n L"l- -, * -] the operator induced by s on the
Fourier transform space-L'[- -, * -] commuteswith multiplication operators
and consequentlyis itself multiplication by a function s(a). Moreover, B(o)
is the boundary value of a function s(p) defined and analytic in the upper
half plane. However, engineersuse, and we shall use, the convention that
s(p) is analytic in the right half plane and that the "Fourier transform" s(a)
of S is a function definedon the imaginary axis.
FUNCTIONS
CHARACTERISTTC
405
Example. Supposethat 9I consistsof a single inductor. Then by definition
u(t; : t(d/dt)i(t) where./ is the inductanceof the inductor. Thus, the scattering
operator S maps
Q*,*r)aa+ Qfi- r),<a
and consequently the scattering matrix for $[ is
S(p) : Qp - l)Qp + 1).-'
The total energydissipatedby any network TL at,a statelu(t), i(t)l : o is
: *"
o, : iI@(o)
.f* uQ)i(t)
(f,Ol+ i(t)l'- lu(t)- r4)f)at.
This is an indefinite quadratic form on the space3Cof states. Every state n'
in the graph of losslessnetworle91 has 0 energy dissipated; 0(n) : 0. Every
state a in the graph of a passiuenetworkhas non-negativeenergy dissipation;
o(n) > 0. A networkis actiueif it is not passive.
Remark 1. Mathematicians began to consider Hilbert spaces with an
indefinite inner product (having infinite dimensional negative part) about
ten years ago. For a survey of the subject see [K]. The example given here
of an indefinite inner product seemsmost natural. In spirit it is a bit like the
one which motivated PhilIip's first paper on the subject [P].
Remarh 2. Actually, what is defined here is usually called a semi-passiue
coincidefor causal,
network. However, the notions of passiueand semi-passiare
(cf.
time invariant networks
VD.
Now we discussn-ports (seeFigure 2). It is a simple variation on the above
analysis. One measuresthe voltage u1(l) across the 1'ft port and the current
iir(l) and irz(t) at the first and secondterminals respectively of the jth port.
Engineersusually assumethat ir'(f) : inQ) : i,(t) and then they work with
the vector valued voltage
["]'l
o(D:l:
I
La"(t))
and vector valued current
["1']
i(D:l:
I
u"(t)J
Everything'that has been said generalizesin an obvious way to vector valued
voltages and curents. The assumption iiLQ) : f;r(l) essentially puts a re-
406
J. W. HELTON
striction on the type of interconnectionsbetween networks which is allowed.
The precedingexposition motivates the following definitions.
Definitions. an n x n scatteringmatrir s(p) is an n X n matrix varued
function analytic on the right half plane. s(p) is called.contrq,ctiue
if. lls(p) ll s t
for p in the right half plane and losslessif s(p) is unitary when p is pure i^uginary. A scattering matrix is called rational if s(p) is a rational function of p.
rt follows directly from the definitions of I and o above that a network $t
is passiveif and only if its scatteringoperator s has ll8ll < 1. This is equivalent
++
trt
v1
tlz
v2
*
r31
n port
in1
vn
1n2
Frcunp 2.
407
CHARACTERISTIC FUNCTIONS
to saying that the scattering matrix 8(p) for gil is contractive. Similarly a
network is losslessif and only if its scattering matrix is lossless.
There are three ways in which networks are connectedtogether: seriesconnection, parallel connection and cascadeloading. We shall discuss cascade
loading. Supposethat Ul[ is a 2 port network and S[ is a 1 port network. SIl is
said to be loaded or cascadedor cascadeIoaded with ft, if one port of :nI is
attached to the port of t'{,.In this processone port of lIlI is left free and the one
port network obtained by using this free port is called the cascadednetwork
(seeFigure 3.)
Let S be the scattering operator for slt. The operator 8, recall, maps an L"-2
vector valued function, thought of as incoming, into an L'-2 vectol valued
function, thought of as outgoing. We write S as a matrix with operator entries:
(1.0)
- fi",l
5
l -l :
lin J
[s,, s,,l [i",] _
I
l&'
ll l:
&,J tm,J
["*'l
LouhJ
Similarly the scattering operator I for 9I maps I in : out. Cascading$1
and Y|^(simply sets outz : in and inr : outr and with simple algebra one can
compute that the scattering operator es(S) is
(1.1)
e'(S) :
S,, + B,,S(1 -
S22S)-IS2,.
Remark 3. Actually there is a subtlety here. Engineerssometimesassume
impticitly that there is an ideal transformer addedto Figure 4 in the connection
between 9I and UIt.This forcesir1(l) : ir"(t).
Everything said above extends in an obvious way to the case where $It-is
a,li + n port and 9t is an ri. port. The only changewhich occurs is that the
inl
""=f*I
ln -L___J
ouEl
Frcunn 3.
408
J. W. IIELTON
functions in1 , in2 , in, outl I out2 I and out must be taken to be vector valued
functions in L". rt is possibleto take Fourier transforms of s and B and then
apply the above argument to calculate that the scattering matrix es(s)(a)
for the cascadednetwork is
(r.2)
es(S)(z) :
S"(a) *
S,,(a)S(e)(1 -
&"(a)S(z))-'S,,(a).
so far, what we have done has been purely algebraic and formal becausein
both (1.2) and (r.1) we assumedthat the inverse of a certain quantity exists.
This may not be the case,but in interesting exampresthe lack of a bounded
inverse causesno difficulty.
Naturally the networks which electrical engineersstudy are electric circuits
whose componentsare, in the caseof passivenetworks, capacitors,inductors,
resistors,ideal transformers,coupled coils, and gyrators. The scattering matrix
for such a network is always a rational function. The scattering matrix also
satisfiess(p) : S(p). The basic abstract fact of passivenetwork realizability
theory is-given a rational n X n scattering matrix there is an electric circuit
9I,whosescatteringmatrix is S(p). The fact that every losslessscatteringmatrix
can be realized by a network 9i,, mathematically speaking,can be gotten from
Theorem vr 3.1 [N-F]. The next section presentsthe mathematical theorem
which engineersuse and then interprets it. Note that there are usually many
networks with the samescattering matrix.
Remarh 4. Instead of associatingthe operator I given by (1.0) with the
2 port JII of Figure 3 engineers[D] sometimesassociatethe 'chain operator' .ll
defined by
.,ft'
[out,
-tl
lrn. l
lo"tJ
with sn. The chain operator naturally is associatedwith an operator entried
matrix
(1.3)
{f:
fe, Bl
[c D)
and.d., B, C and D are easy to compute in terms of S11, Sr, , 51 and Srz . The
... I uIIp
advantageof thisformalismisthatwhen one connects2ports {rL1 75(L2y
together in a long chain to form another 2 port Jr[ the 'chain operator' for sl"d
is just the product .Il1 . .. 1l2.ll, of the chain operators 11.;for JII; . The disadvantage of this formalism is that 1t is not unitary for lossless $tI. However,
'll does satisfy clt/'U* : .,I and .U.*,.I1[:
..I where ./ : G _f). Also, tt is bounded
if and onlv if lls"ll ( I and lls,,ll < 1. The operator €s(s) defined immediately
above equation (1.f) can be expressed in terms of .lt as
(1.4)
s"(s): e,(s): (/s + B)(cs+ D)-,
CEARACTERISTICFUNCTIONS
409
'general symplectic'
Maps of the form (1.4) where 'u"/u* : J are called
and they have been invesiigated independently by the mathematicianspniltips
[Pl, Krein-Smul'jan [K-S] and others. Work on such maps is an important
part of the study of Hilbert spaceswith an indefinite inner product (mentioned
in Remark 1). Also, the 1-1 conespondencebetween ,,I-unitary and unitary
operatorsextendsformally to a correspondencebetween J-expansive operators
(possibly unbounded) and contraction operators. The ,.f-expansiveoperators
are discussedin [K-S] as a generalization of "I-unitary operators.
The realizability of a scattering matrix. Let H be a complex Hilbert space
with norm ll ll and let s(H) denote the space of bounded operators on H.
Any operatorin S(//) with norm lessthan or equal to oneis calleda contraction.
we shall assumelhat H is the complexification (seechapter 9, section 2, [sl)
of a real subspacert of.H. This determinesan operation which we denote by
x -+ fr on Il which we call complexconjugation. The conjugateA of an operator
,4,in "c(I/) is definedby Ax : E f.o, all.x e H. An operator A n s(H) is called
real if A : A. A corollary of Theorem VI 3.1 [N-F] is
Theorcm R. 4 S(p) is a contra,ctionaoJuedfunction analgtiri'n the right
hatl planewiththeproperties
lls(t)oll < llrll if xeH and S(F): S(p),thmthere
is a Hilbert spa,c,eHand operolors
Sr1 :.EI -+ Il,
Sr2 : E[ + Il,
8r, : II -+ H,
S22 : E[ -r E,
where
s: [s" s"l
Srr)
[S'
is a u,nitarymap of H @ H ontoH @H suchthat
(2.0)
S(p) :
Srr *
Srze(l -
Srra)-'S'
and.z : (p - l)/(p I L). Moreouer,thereis a complexconjugation- orlEt
suchthat 8 is reaLwith respectro - @ - on H @H. This conjugationis explicitly
construrted.
Proof. Let 0(z) : S(p). Theorem VI, 3.1 [N-F] says that there is a Hilbert
spaceII and a contraction operator T on H such that d(a) coincideswith the
characteristicfunction 0{z) of.T. By definition (seeChapter VI, (1.1) tN-Fl)
(2.r)
|aQ) : [-T + zD7.(I - aT*)-'D1] l'.
where I denotesrestriction of an operator to a subspaceand where
Dr. : (f - 1A:*;'z',
D, : (I - T*T)'/',
Dr : DrH
a,nd D1. : Dr.If.
410
J. W. HELTON
For each e the operator 042) maps o1 into o". , a facr which follows from the
identity TDr : DpT (see[N-F] r, (3.4)). The statementthat dr and g coincide
(see Chapter Y, (2.4) tN-Fl) me&ns that there are unitary maps U and.V
U: D7. --+H,
V:H-+9,
such that
(2.2)
o(r) : UorQ)V.
Let P(P*) denotethe orthogonalprojection of EI onto Dr and oa. respectively.
when one combines (2.1) and (2.2) one gets a representationfor 0(z) o;r.d,
consequentlyfor S(p) of the form desiredprovided that
s : i- uP*Tv uP*Dr,)
r* )
lorv
is a unitary map of l/ @ H onto f/ @ H.
Now we prove that E is unitary. Supposethat 6) 11 @ H. Then a simple
"
computation along with the fact lhal U is unitary gives
us that
ll /-\ ll,
llt$/il
- (P*TVu,P*D,,a)
: (P*TVx,P*TVr)
- (P*Da.!,P*TVa)
* (P*D7.y,P*Dr.A) * (DrVr, DrVr) * (DrVr, T*y) + (T*A, DrVx)
I (T*y, T*a).
The definition of D1 and D1. , the identities TD1 : DpT and D1T* : T*D1. 1
and the consequence TP : P*T of these identities, when used with the above
formula yield
: ll(rll'
+tat',
rrcn,
ll'(;)ll':
Thus E is an isometry. Next we prove that the range of E is densein I/ @ H.
Supposethat
:'
'()(i.)
forall ueH
and yeII. Then
-(UP*TVu,t) + (UP*Dpy,l) * (DrVr,w) t
We seefrom setting z : 0 that
(2.3)
(2.4)
Dr,(J-tl * Tar : o
and from setting A : O lhat
-V-|P(T*U-'.C (2.5)
Drul : g.
(T*y,w):
g.
CHARACTERISTIC FIINCTIONS
41r
Since T*Dr. C Dr , the vector m : T*U-'l - Drw belongsto Dr and so
V-'m :0 implies m : 0. Now multiply Q.$ by Dr. , multiply mby T, subtract
to obtain U-'l :0. Thus / : 0 and equation (2.3) reducesto
the two expressions
( D ' 1 V u , w *) ( T * a ,? o ) : 0
for all r and y. Consequently
D7w :0
and Tta : 0,
that is, w - T*Tw : D?iLl : 0 and T*Tu : 0. Therefore w : 0 and we have
proved that the range of the isometry E is densei" fl O H. ConsequentlyE is
unitary and the first part of Theorem R is proved.
Now we construct a conjugationon.EIG) H x'ith respectto which the operator
E is real. This part of the proof is not self contained,since to make it self contained would require the reconstruction of several large parts of the book
[N-F]. Thus we assumehere that the reader has a good knowledgeof the proof
of TheoremVII, 3.1 [N-F], a knowledgeof conjugationoperators(seeChapter 9,
Section 2 [5]), and we use the notation of [N-F] without first introducing it.
Let g denote the conjugation operation induced on L'QI) by -; namely
h(e) :
*hor'o'
4
ino"'r'
It is clear that the subspacesH"(H) and the orthogonal complement of.H2(H)
arginvariant under ;t. The spaceH'(H) is invariant under 0 and the hypothesis
0(z) : o(e)says preciselythat
g0 : 09.
This implies that the operator I given by multiplication bV A(t)
II - 0(er')*0("")J"" on L'(H) commuteswith J. We mention at this point that
an operator is real (with respect to 5) if and only if the operator commutes
with J. One can see from equation VI (3.29) [N-F] that the Hilbert spaceE
which appearcin the statement of Theorem R is
-"
H : lH'(rI @E(n"l
Q lor@ l0: w eH'(H)l
denotes closure. Define ! to Ue the conjugation operator on K =
where
definedby
L'(H) @E@
'What
3@.@ u) : $u*@ gu for z* @ u e K.
we have done above implies that II and flr are invariant under j. Thus
j commuteswith the orthogonalprojection Pn of K onto H.
The transformation U of K onto itself defined by
U(r* @ u) : e"u*@
"t'u
for z* @ u e K is, accordingto Theorem VI (3.1) [N-F], the minimum unitary
dilation of T. That is,
4r2
J. W. EELTON
T : psUls.
Clearly U commuteswith 3, and the computation
lr : e.ivls : PsUSl' : PgUlr 7i : tS
shows that ! commuteswith T. Henceforth, let us write the conjugation map
on H induced bV J as t -> f. From the fact that T : f we may conclude that
the operator S : II @ II -+ I{ @ H defined by
(
-r
(f - tl*;'zr1
s: |
- T*1",
[1^r
T*
|
)
is-@-real.
The proof of Theorem R would be completeif we could show that
tlfr:Ur
and Vg:fr
for every z e D1. ar..dy e I/. The explicit construction of the map 7 is found
by combining equation VI (3.24) [N-F] which exhibits an isometry w of. H
onto L as
w:A--+W*
LY
for YeH
with equation II (1.6) [N-F] which exhibits an isometry ,p of.L onto D.1 as
e(J - T)h: Drh for heE.
The isometry V rs 9w.If y e fl, then
og+ Lg : ofia* Lfla : Aloa* Lyl,
where y is thought of both as being in fI and as being a constant function in
a'(I/). Since
jo, : nr|
and Sg - T) : (U - T)g
one gets that V0 : |VA : VA.The isometry U is treated similarly.
fn TheoremR we assumedthat ll8(t)zll < llrll for all r tn H. Suchanalytic
functions are called purely contractive.
Proposition V (2.1) of [N-F] says that
Theorem D. 4 S(p) is a contra,ctionaaluedanalytic function, thm H con
into the orthogonaldirect sum of two subspacesH : H, @ H,
be decomposeil
in eucha way that
r
(1) S(p)H, ( Hi for aIIP"ep ) 0 an4 i : 1,2;
)
(2)
anil S(p) on H 2is a constantunitary oTterator.
S(p) on H 1is purely contractiue
In the case where I/ is finite dimensional and s(p) is losslessand rational,
Theorems D and R taken together are precisely Theorem 3.3 in nfl. In fact,
the dimension of II in the Nagy-Foias construction of Theorem R seemsto be
CHARACTERISTIC FUNCTIONS
418
the electrical engineering "Smith-McMillan" degree (c.f. Cor. 3.2 in [W]),
just as one would expect from Theorem 3.3 [W]. However, when S(p) is not
losslessthe correspondencebetween the electrical engineering approach and
the Nagy-Foias approach breaks down. With the Nagy-Foias approach one
needs an infinite dimensionalEI to representany "lossy" passives(p). Physically this calls for a circuit with infinitely many components-somethi.g which
is naturally anathema.
Remark 5. Engineers, when treating "lossy" networks, represent S(p)
as 0r((p - t)/(p + 1)P) where P is someorthogonal projection onto II. This
extra freedomallowsthem to represent8(p) with aII which is finite dimensional.
fn order to describethe connectionbetweenTheoremsD and R and network
realizability theory, we need to know two things. First of all, it is at least in
theory possiblefor engineersto build a circuit which realizesa given constant,
losslessscattering matrix (S(p) : constant').The fact that constant lossless
scattering matrices are mathematically so simple, translates into the physical
fact that they can always be realizedby a circuit which consistsonly of transformers, gyrators, and wires interconnectingthe ports. The secondthing that
we need is the fact that the scattering matrix for an inductor is S(p) :
(p - t)/(p f 1) (recallthe example).
For the sake of this presentation we shall take losslessconstant scattering
matrices to be basic and discussthem no further. Thus Theorem D reduces
the realization problem for an arbitrary S(p) to the purely contractive case.
supposethat one wants to build a circuit which realizesa given purely contractive n X n scattering matrix S(p) and that he has calculatedthe matrix S
which appearsin Theorem R. To solve the problem one first builds lhe n * m
port Ul(which has E as its scattering matrix (seeFigure 2). Here zr,is the dimension of H. Acrosseach of the last zn ports (ports associatedwith I{) one places
a unit inductor. The scattering matrix for this circuit is seenfrom (1.2) to be
precisely the one given by (2.0). Thus we have constructed a circuit whose
scattering matrix is precisely the scattering matrix S(p) that we wanted to
synthesize.
Appendix. The network realizability theorem which is discussedin this
article is actually a manifestationof a basictheoremin systemstheory. Namely,
Theorem. Euery matrix aalued,rational function G(s) which is onalytic in
the unit d:iskcan be written in the f orrn
(A.1)
G("): D*zC(I
-zA)-tB
whereA, B, C, D arefi'nite ilimensionalmatricesan'd'lo(A)l < l.
Facts of this type were known to mathematieians since the mid-fifties. An
elegant proof of this theorem which is suitable for computer implementation,
wa.sgiven by systemstheorists B. L. Ho and independently by Youla and Tissi.
414
J. W. EELTON
rt is interesting to compare their proof with the proof of the closely related
Theorem VI 3.1 [N-F]. If one expandsan inner function G(a) n a power series
about the origin and appliesthe Ho construction (cf. Theorem 6.3 [C] or Ch. 10,
Sec. 11 IA-F-K]) one can trace the correspondencebetween it and the N-F
construction. rn fact, the rro and N-F constructions of the basic operator :4.
in representation(A.1) are equivalent.
R. Kalman has proposed[A-F-K] that a certain module theoretic viewpoint
gives the best approach to this branch of systemstheory. Independently, two
operator theorists B. Moore and E. Nordgren IM-NI tNl in studying NugyFoias canonical models have taken a module theoretic viewpoint very similar
to the one proposedby Kalman.
Rrrpnrttcns
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[D] P. DnWtmn, Roomy scatteringmntria sgntheses,
Technical report, Ifniversity of California,
Berkeley, 1971.
tlq M. G. KmtN, Introiluction to the geometryof ind,efinite J-spacesanil to the theorg ol operolors
in thosespoces,Amer. Math, Soc. Transl. 2, 93 (1970), 108-176.
[K-S] M. G. Krein & J. Smul'jan, Fronti'onallinear transformations with operator cofficients
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(Russian).
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15-27.
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Amsterdam, 1970.
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(1970), 467-488.
This research was partially supported by N.S.F. Grant GP-19582.
State University of New York
Stony Brook
Date com.mnnhaled.'Novsuann 12, lgTL
a