- Email: [email protected]
Theory and applications of -lossless scattering systems
Theory and Applications of J-lossless Scattering Systems? 6)
N.
LEVAN
School of Engineering and Applied Science Universit_yof California, Los Angeles, California
ABSTRACT:
We introduce
the concept
of J-scattering
operators
and J-lossles~
scattering
operators on a Hilbert space with the aim of using them for the cascade load synthesis of a class of active operators. It is shown that the active operator8 can be synthesized by cascade loading their J-lossless dilation in unit resistors. Also a second type of dilation is constructed for the synthesis of general active operators.
I. Introduction
In this paper, the properties and applications of a new class of lossless scattering systems (on a Hilbert space) called a J-lossless system are studied. Heuristically, a lossleas system is one which is passive and which does not absorb energy delivered to it from the outside world. In other words, for a system to be lossless, its input energy has to be equal to its output energy. Interpreting energy as the square of the norm on a Hilbert space, we see that a lossless system is characterized mathematically by an isometric operator on the space: in particular, if the operator maps the input space onto the output space, then the isometry is a unitary. In a similar way, a J-lossless scattering system is one which does not absorb “J-energy” delivered to it. The “J-energy” is defined by means of a new inner product defined for the Hilbert space. This inner product can be interpreted as the cross-correlation between the input and output of a special J-system. In Sect. II key facts about scattering systems (on a Hilbert space) are discussed. These include the concept of passivity, activity and losslessness. Section III is devoted to the notion of J-spaces and operators on these spaces. We introduce here the definition of a J-scattering operator and the concept of passivity, activity and losslessness for these operators. Section IV is concerned with the synthesis of active scattering operators by means of J-lossless systems. We review here, briefly, the cascade load procedure for realizing a passive scattering operator. It becomes evident that both passive and active operators can be realized by the cascade load procedure, which is shown to be a particular case of a general analytical method for studying the operators called the characteristic operator function method. t This work was supported by N.S.F.
Grant GK-23982.
313
N. Levan ZZ. Scattering
Systems
and
Scattering
Operators
on Hilbert
Spaces
In this paper, unless otherwise stated, a real separable Hilbert space is always denoted by H. The inner product and norm on H are denoted by ( , ) and 11)Irespectively. Some of the key results of linear scattering systems are given as follows. A system 9’ is said to have a scattering description if it is characterized by an operator S, called “the scattering operator”, mapping incident waves (inputs) x into reflected waves (outputs) y (1). Here, we take x and y to be vectors of a Hilbert space H, hence S is a map from H + H : y = Sx,
x,y~H.
(1)
In particular, we consider only linear operators S on H. Since the system 9 is characterized by its scattering operator S, we shall, in what follows, either speak of properties of 9 or those of S. A linear 9 is said to be “passive” (weakly) if .Q = I[x(~~--~I~SX~~~>O,ZEH. This is clearly equivalent
(2%)
to either one of the following: (2b)
IISIIG 17 I-iPS~0,
(2c)
where in (2c), S* is the adjoint of S. Condition (2b) means that a passive S is a “contraction” operator on H (2), while (2~) implies that when S is passive, I-S* S is a positive operator. Let 11 S IIbe the norm of S, then, since 11 S 11= I(S* I/, we have for a passive 9, the following equivalent conditions : es. = ~[x~)~-~~S*X~~~~O, XEH,
(3%)
WI
IIs*Il G 1 and I-SS”
2 0.
(34
From the above, we obtain: Proposition 1. A linear Y is passive if and only if the following equivalent conditions hold : (a) E& 0 and es* > 0, (b) S and S* are contraction (c) I-Sx*S>O
operators on H,
and I-SS*>,O.
A linear passive 9’ is said to be “lossless” H and Q = 0,
if S is passive and maps H into
VXEH.
Pa)
From which it follows that s*s=I,
314
(4b)
Journal of The Franklin Institute
Theory and Applications
of J-Lossless Scattering Systems
i.e. a lossless S is an “isometry” on H. An isometry H(SH = H) is called “unitary”, and we have
which maps H onto
s*s=ss*=I*
(5)
If 9’ is not passive, it is called “active”. For our purpose, generally called “non-passive”, and we define Definition 1. DeJinition 1. A linear bounded scattering active if 11 S II> 1, and “doubly active” if cs = j~~/~~-~~Sx~]~
operator
such an S is
S is defined
XEH,
to be (64
and cs, = IIx/~~-IIS*X/~~
XEH.
that, for a doubly
(6b)
active S:
(7)
l
active
implies active.
Proposition 2. A linear Y equivalent conditions hold :
Furthermore,
is doubly
when only ss< 0, S is
active if and only if the following
(a) cs < 0 and ss. -C0, (b) S*S-I>0 III.
and SS*-I>O.
The J-Scattering
Operators
We review, in this section, some of the main facts of the so-called J-spaces, and operators on these spaces, as introduced and studied by Phillips (3), Krein (a), Ginzburg (5) and others. Let H, as before, be a real separable Hilbert space, and let J be a linear bounded operator from H + H such that J=
J*
By means of J, & new inner product,
and
J2 = I.
(8)
denoted by ( , )J can be defined on H:
(X>Y).7 = (JGY) = (G JY),
x,y~H.
The pair [H( , )=I is called a J-space and is denoted define Definition 2.
(9) by J-H.
We now
Dejinition 2. A scattering operator S on a J-space is called a “J-scattering operator”. A linear bounded operator T on J-H is said to be J-passive (or J-contractive) if Ed = (x,+-(Tx,T&>O, VXEH. (10) This equation can be written as sj = (Jx, 2) - (JTx, TX) > 0 = ([J-T*JT]x,x)>O,
Vol. 294, No. 5, November
1972
(11) XEH.
(12)
315
N. Levan Hence (10) is equivalent
to J-T*JT>O.
(13)
We note that, unlike contractions, if T is J-contractive, it does not mean that T* is also J-contractive. This is quite clear from (11). If both T and T* are J-contractive, i.e. J-T”JT>O and J-TJT*>O
(14)
then T is said to be “J-bicontractive”. If T is not J-contractive, it is said to be “J-non-contractive” (or “J-active”). A J-contractive operator T is said to be “J-isometric” (or “J-lossless”) if ej = (cc,&-(Tx,T&
= 0,
XEH,
(15)
or equivalently T”JT
= J.
(16)
A J-isometry T which maps H onto H(TH = H) is called a “J-unitary” operator. Thus for a J-unitary operator we have T” JT = TJT* = J.
(17)
Further properties of J-spaces and J-operators will be introduced as needed in subsequent sections. We conclude this section with a couple of examples of operators on H and J-H. Consider the operator
_I]
J=[f
(18)
on the product space H x H. It is clear that J satisfies all the properties in (8). We thus have the J-space
[f
On HxH,
-HxH.
-I]
let S be the
matrix operator S
s=
[ -Szl
(19)
.‘” I ’
where, for the moment, S,, and S2i are bounded linear operators from H to H. We have for S: e&y =
Ijxl12-IIsx/(2,
x =
;; [
= IIXl II2+ II52 II2 - ( IIs2151 and for J-S
= (II% 112-
II fJ2,~ll12)
E~-~ = (Jx,z) - (JSx, Xx),
1
(204
(xI,x~EH)
II2+ II42
x2
II”)
+ ( II x2112- II S,2x2112)
x =
L
z;
I
316
IIs,,x,
/I2 - ( IIx2 II2 + I! &2x2
POc)
(~1, x2 E H)
(2W
= II”1112-ll~2112-(llfG2~2112-lI~21~1112) = II Xl II2 +
Wb)
Wb) II”).
Journal of The Franklin
WC)
Institute
Theory and Applications
of J-Lossless Stuttering Systems
It is clear from (20~) that for S to be passive, both S,, and S,, would have to be passive. Also from (21~) it is evident that, for this example, the passivity of S does not imply the J-passivity of J - S and vice versa. Even when S is lossless, i.e. when AS’,,= S,, = I, J-S need not be J-lossless. Finally, consider on H x H, the simple operator s=[S;
I],
(22)
we have es = 11~~1/2-11~11~~112+1152112~
(23)
~13~2~H7
and &J--S= 11~~112-11~~~5~112-11~2112,
x~~x~EH.
(24)
Thus if S,, is lossless on H, S is passive on H x H while J -S is J-active on J-HxH. A simple example of an S which is lossless and J-lossless at the same time is when
s= [ “rl s,,l’
(25)
where both S,, and S2a are lossless on H.
IV.
Synthesis
of Doubly
Active
Scattering
Operators
on a Hilbert
Space
We now discuss the problem of synthesizing a given doubly linear actve scattering operator S on a Hilbert space. It is shown that “any doubly active S on H can be realized by cascade loading its J-lossless dilation on H x H in unit resistors”. This result is, clearly the exact analog of the cascade load synthesis procedure for realizing passive scattering operators on H (6). To exhibit the analogy between. the synthesis of passive and active operators, we briefly consider first the cascade load synthesis of passive operators. Let S, for the moment, be a linear bounded scattering operator from H to H. A dilation (7) of S on H x H is defined to be the matrix operator x = [ ;1
Z:: 1)
(26)
where Xx2, X2, and X22 are linear bounded operators from H to H. Cascading C in an operator St( : H -+ H) as shown in Fig. 1, we have for X : y = cx,
y = [;;I,
cz = E;],
q,52,~1,~2~H,
(27)
or
Vol. 294. No. 5, November 18
1972
Y1=
Sx,+~,,x2
(28)
Y2 =
~21x1
(29)
+x22x2
317
N. Levan and for S,: (30)
x2 = S,Y,* Eliminating
x2 and yz in the above, we find (31)
provided, of course, that the inverse on the right-hand side exists. Thus, it follows that the scattering operator S, of the cascade combination “2 - Si’ (if it exists) is s, = s + Xc,, S,(I - Es2 X,)-l c,,. (32) (1) Now if S is passive, it can be realized by requiring Z to be lossless, and by cascade loading C in unit resistors which corresponds to S, = 0. One such lossless C is the unitary dilation (7) of S given by
Z=
S D, (= [I-S*S-J*) [
-D,*
I*
(= - [I-ss*]* S”
(33)
We have shown” that the cascade load synthesis of a passive S is closely related to the “characteristic operator function (C.O.F.)” (8) of the contraction (i.e. passive) S on H. Indeed by choosing I; to be the unitary dilation (33) and S, = AI, where X is a complex scalar, we have from (32) S,(h) = S -D,.
A(1 - AS*)-l D,,
(34)
where for A(1 - AS*)-1 = (X-1 - S*)-l t o exist, A-l has to belong to the resolvent set p(S*) of S*. Now since X is lossless, the passivity of S,(h) depends only on that of S,(h) which is clearly passive for 1h 1-c 1. Furthermore, since S* is also passive, p(S*) contains the set {A: ]A] < l}. Hence, we conclude that S,(X) of (34) exists and is passive for 1X ( < 1. The passive operator S,(h) when restricted to R(D,): the closure of the range of D, and regarded as an operator from R(D,) to R(D,,) is called the “characteristic operator function” 19,(x) (8) of S: B,(h) = [S-D,,h(I--XS”)D,]IR(D,),
(h]
(35)
(2) Consider now the case of doubly active S. It is clear that the dilation Z of Eq. (33) is no more defined in this case since I-S* S and I - SS* axe now negative operators. Instead, we consider the dilation 2 of S defined by f;=
S A, (= [s*S-I]*) C
* N. Levan: Submitted for publication.
318
A,, ( = [ss* - I]f) Sl”
1,
Joumal of
The Franklin
(36)
Institute
Theory and Applications
of J-Lossless Scattering Systems
where A, However,
and A,, are now well defined, due to Proposition 2, Section II. 2 is no more an unitary operator on H x H, but instead, it is a 1 . J-unitary operator on the J-space -Hx H.* We call 2 the -1 [. “J-unitary dilation” or “J-1ossless dilation” of S. 1 . Theorem I. The -unitary dilation 2 of doubly active S is an -1 L I active operator on H x H.
1
We first show that for 2
Proof:
E=IIxlj2-~~y~~2<0,
x=
,
[1
y=Ec=
;:
(2, YEH x H)
= ll~Il12+llx2112-(llY1112+IIY2112)*
NowsinoeCisJ=[:
(37) (38)
_1]-lossless,wehaveforeJofg E., = (JGx)-(JY,Y)
= 0,
where x and y are as defined in (37). Hence 11%112-11~2112-(IIY1112-IIY2112)
or
= 0
IIx1112-/IY1112
Therefore,
e = 2( II Xl /I2 - II Yl
= 11~2112-IIY2112*
112) = 2( IIx2II2 -
I
(39)
II Y2 l12h
Now y1 = Sz, + A,. x2.
Thus,
4~ = ll~1112-II~~~+As~~2112 = ~Il~,II+II~~,+~,~~,II~~II~,II-II~~,+~,~~,Il~~
(40)
The second term in ( ) on the right-hand side is always negative since both S and A,, are active. Similarly, for the expression of E in terms of I(x2 11and IIyz 11. We conclude that E is negative and therefore f; is active and the theorem is proved. Cascade loading 2 in S, = XI, we have from (31) and (36) : y1 = [S + A&(1
- XS*)-l A,,] x1,
therefore S,(X) = S+A4,h(1-XX*)-iAs*,
(41) (42)
where, as before, h-lsp(S*), which in this case contains the unit disc D, of radius r < 11 S* II-1< 1. Thus we obtain Theorem II. Theorem II. The scattering operator S,(A) = S+AsX(I-XX*)-lAs,
AED,,
143)
obtained by cascade loading the J-lossless dilation 2 of an active S in hl is an active scattering operator on H. * N. Levan: Submitted for publication.
Vol. 294, No. 5, November
1972
319
N. Levan Proof: we have for s, E = II~~l12-IIy~l12~ +Y~EH>
(44)
E = ~ll”~ll2+II~2ll2-(IIY~ll2+IIY2ll2)1+(lIY2/l2-Il~2ll2)~
(45)
where x1, x2, y2, y2 are, as in the passive case, related by (due t’o the cascade combination 2 - XI) Yl = S,(X)
(46)
x17
(47) and x2 = hy,.
(48)
Now, the term in { } on the right-hand side of (45) is just the quantity E of 2. Since 2 is J-lossless, we have, by using (39), E = 2( II x2 /I2 - II Y2112) + ( Ii Y2II2-
IIx2l12).
therefore E = IIx2112-llY2112 = (I~I-l)llY2112~ where use has been made of (48). Hence E
for Ih/< 1
and the theorem is proved. We note from the proof of the theorem that S,(X) would be passive for IX ( > 1, as long as it exists, i.e. as long as p(S*) contains some domain outside of the unit disc. It is clear from (43) that S,(O) is just S, we have Theorem III. Theorem
III.
Any
doubly
active
scattering
operator
S on H can be
realized by cascade loading its
-1ossless dilation in unit resistors. [t -11 Note that, by restricting S,(h) on R(A,), we can also define the C.O.F. for a linear doubly active operator S on a Hilbert space. Finally, it is clear that if S is active, then S/m, where 1~ IIS/I
film Slm= [ [I - (S” S/na2)]f =-
-
[I - (ss*/m2)]*
S*lW&
ia [ (m2A*s)’
-(rn2I-SS”)* s*
1
(49)
1*
Cascade loading Eslm in XI, we find S,(h) =
320
~-(m21ss*)t~(l-~~)-1(~21-s*s)~, IhI< 1. Journalof
The Franklin
(50)
Institute
Theory and Applications
of J-Lossless Scattering Systems
Thus S,(O) realizes, in this case, the passive operator S/m. Consequently, S can be realized from S/m by the scheme of Fig. 2 (9). CIRCULATOR
m+l -m_l
1.Q
FIG. 1.
Conclusion
We have presented the concept of J-lossless scattering operators on a Hilbert space. One immediate application of these operators is to the synthesis of doubly active scattering operators. It is shown that a given doubly active operator can always be realized by cascade loading its J-lossless dilation in unit resistors. This, of course, is similar to the usual method for realizing a passive operator. Also, it is shown that any S with 11 S II> 1 can be realized from the passive operator S/m where m is an arbitrary constant such that 1 < jlS I/ cm. This realization involves the use of circulators. References
(1) R. W. Newcomb,
“Linear Multiport Synthesis”, McGraw-Hill, New York, 1966. (2) Sz-Nagy, “Extensions of linear transformations in Hilbert space which extend beyond this space”, Appendix to F. Riesz and Sz-Nagy, “Functional Analysis”, Frederick Ungar, New York, 1960. (3) R. S. Phillips, “Dissipative operators and hyperbolic systems of partial differential equations”, Trans. Amer. Math. SOL, Vol. 90, pp. 193-254, 1959. (4) M. G. Krein, “Introduction to the geometry of indefinite J-spaces and to the theory of operators in those spaces”, Amer. Math. Sot. Transl. (2), Vol. 93, pp. 103-176, 1970. (5) J. P. Ginzburg, “On J-contractive operator functions”, Pokl. Akad. Nauk SSSR, Vol. 117, pp. 171-173 (in Russian), 1957. (6) R. Se&s, “Synthesis of general linear networks”, SIAM J. Appl. Math., Vol. 16, No. 5, pp. 924-930, Sept. 1968. (7) P. R. Halmos, “Normal dilations and extensions of operators”, Summa Brad Math., Vol. 2, pp. 124-135, 1950. (8) Sz-Nagy and C. Fioas, “Anrtlyse Harmonique des Op&&eurs de 1’Espace de Hilbert”, Masson et Cie. and Akademiai Kiado, 1967. (9) N. Levan, “Synthesis operators”.
Vol. 294, No. 5, November
1972
of active scattering operators by its m-l-derived
passive
321
N.
LEVAN
School of Engineering and Applied Science Universit_yof California, Los Angeles, California
ABSTRACT:
We introduce
the concept
of J-scattering
operators
and J-lossles~
scattering
operators on a Hilbert space with the aim of using them for the cascade load synthesis of a class of active operators. It is shown that the active operator8 can be synthesized by cascade loading their J-lossless dilation in unit resistors. Also a second type of dilation is constructed for the synthesis of general active operators.
I. Introduction
In this paper, the properties and applications of a new class of lossless scattering systems (on a Hilbert space) called a J-lossless system are studied. Heuristically, a lossleas system is one which is passive and which does not absorb energy delivered to it from the outside world. In other words, for a system to be lossless, its input energy has to be equal to its output energy. Interpreting energy as the square of the norm on a Hilbert space, we see that a lossless system is characterized mathematically by an isometric operator on the space: in particular, if the operator maps the input space onto the output space, then the isometry is a unitary. In a similar way, a J-lossless scattering system is one which does not absorb “J-energy” delivered to it. The “J-energy” is defined by means of a new inner product defined for the Hilbert space. This inner product can be interpreted as the cross-correlation between the input and output of a special J-system. In Sect. II key facts about scattering systems (on a Hilbert space) are discussed. These include the concept of passivity, activity and losslessness. Section III is devoted to the notion of J-spaces and operators on these spaces. We introduce here the definition of a J-scattering operator and the concept of passivity, activity and losslessness for these operators. Section IV is concerned with the synthesis of active scattering operators by means of J-lossless systems. We review here, briefly, the cascade load procedure for realizing a passive scattering operator. It becomes evident that both passive and active operators can be realized by the cascade load procedure, which is shown to be a particular case of a general analytical method for studying the operators called the characteristic operator function method. t This work was supported by N.S.F.
Grant GK-23982.
313
N. Levan ZZ. Scattering
Systems
and
Scattering
Operators
on Hilbert
Spaces
In this paper, unless otherwise stated, a real separable Hilbert space is always denoted by H. The inner product and norm on H are denoted by ( , ) and 11)Irespectively. Some of the key results of linear scattering systems are given as follows. A system 9’ is said to have a scattering description if it is characterized by an operator S, called “the scattering operator”, mapping incident waves (inputs) x into reflected waves (outputs) y (1). Here, we take x and y to be vectors of a Hilbert space H, hence S is a map from H + H : y = Sx,
x,y~H.
(1)
In particular, we consider only linear operators S on H. Since the system 9 is characterized by its scattering operator S, we shall, in what follows, either speak of properties of 9 or those of S. A linear 9 is said to be “passive” (weakly) if .Q = I[x(~~--~I~SX~~~>O,ZEH. This is clearly equivalent
(2%)
to either one of the following: (2b)
IISIIG 17 I-iPS~0,
(2c)
where in (2c), S* is the adjoint of S. Condition (2b) means that a passive S is a “contraction” operator on H (2), while (2~) implies that when S is passive, I-S* S is a positive operator. Let 11 S IIbe the norm of S, then, since 11 S 11= I(S* I/, we have for a passive 9, the following equivalent conditions : es. = ~[x~)~-~~S*X~~~~O, XEH,
(3%)
WI
IIs*Il G 1 and I-SS”
2 0.
(34
From the above, we obtain: Proposition 1. A linear Y is passive if and only if the following equivalent conditions hold : (a) E& 0 and es* > 0, (b) S and S* are contraction (c) I-Sx*S>O
operators on H,
and I-SS*>,O.
A linear passive 9’ is said to be “lossless” H and Q = 0,
if S is passive and maps H into
VXEH.
Pa)
From which it follows that s*s=I,
314
(4b)
Journal of The Franklin Institute
Theory and Applications
of J-Lossless Scattering Systems
i.e. a lossless S is an “isometry” on H. An isometry H(SH = H) is called “unitary”, and we have
which maps H onto
s*s=ss*=I*
(5)
If 9’ is not passive, it is called “active”. For our purpose, generally called “non-passive”, and we define Definition 1. DeJinition 1. A linear bounded scattering active if 11 S II> 1, and “doubly active” if cs = j~~/~~-~~Sx~]~
operator
such an S is
S is defined
XEH,
to be (64
and cs, = IIx/~~-IIS*X/~~
XEH.
that, for a doubly
(6b)
active S:
(7)
l
active
implies active.
Proposition 2. A linear Y equivalent conditions hold :
Furthermore,
is doubly
when only ss< 0, S is
active if and only if the following
(a) cs < 0 and ss. -C0, (b) S*S-I>0 III.
and SS*-I>O.
The J-Scattering
Operators
We review, in this section, some of the main facts of the so-called J-spaces, and operators on these spaces, as introduced and studied by Phillips (3), Krein (a), Ginzburg (5) and others. Let H, as before, be a real separable Hilbert space, and let J be a linear bounded operator from H + H such that J=
J*
By means of J, & new inner product,
and
J2 = I.
(8)
denoted by ( , )J can be defined on H:
(X>Y).7 = (JGY) = (G JY),
x,y~H.
The pair [H( , )=I is called a J-space and is denoted define Definition 2.
(9) by J-H.
We now
Dejinition 2. A scattering operator S on a J-space is called a “J-scattering operator”. A linear bounded operator T on J-H is said to be J-passive (or J-contractive) if Ed = (x,+-(Tx,T&>O, VXEH. (10) This equation can be written as sj = (Jx, 2) - (JTx, TX) > 0 = ([J-T*JT]x,x)>O,
Vol. 294, No. 5, November
1972
(11) XEH.
(12)
315
N. Levan Hence (10) is equivalent
to J-T*JT>O.
(13)
We note that, unlike contractions, if T is J-contractive, it does not mean that T* is also J-contractive. This is quite clear from (11). If both T and T* are J-contractive, i.e. J-T”JT>O and J-TJT*>O
(14)
then T is said to be “J-bicontractive”. If T is not J-contractive, it is said to be “J-non-contractive” (or “J-active”). A J-contractive operator T is said to be “J-isometric” (or “J-lossless”) if ej = (cc,&-(Tx,T&
= 0,
XEH,
(15)
or equivalently T”JT
= J.
(16)
A J-isometry T which maps H onto H(TH = H) is called a “J-unitary” operator. Thus for a J-unitary operator we have T” JT = TJT* = J.
(17)
Further properties of J-spaces and J-operators will be introduced as needed in subsequent sections. We conclude this section with a couple of examples of operators on H and J-H. Consider the operator
_I]
J=[f
(18)
on the product space H x H. It is clear that J satisfies all the properties in (8). We thus have the J-space
[f
On HxH,
-HxH.
-I]
let S be the
matrix operator S
s=
[ -Szl
(19)
.‘” I ’
where, for the moment, S,, and S2i are bounded linear operators from H to H. We have for S: e&y =
Ijxl12-IIsx/(2,
x =
;; [
= IIXl II2+ II52 II2 - ( IIs2151 and for J-S
= (II% 112-
II fJ2,~ll12)
E~-~ = (Jx,z) - (JSx, Xx),
1
(204
(xI,x~EH)
II2+ II42
x2
II”)
+ ( II x2112- II S,2x2112)
x =
L
z;
I
316
IIs,,x,
/I2 - ( IIx2 II2 + I! &2x2
POc)
(~1, x2 E H)
(2W
= II”1112-ll~2112-(llfG2~2112-lI~21~1112) = II Xl II2 +
Wb)
Wb) II”).
Journal of The Franklin
WC)
Institute
Theory and Applications
of J-Lossless Stuttering Systems
It is clear from (20~) that for S to be passive, both S,, and S,, would have to be passive. Also from (21~) it is evident that, for this example, the passivity of S does not imply the J-passivity of J - S and vice versa. Even when S is lossless, i.e. when AS’,,= S,, = I, J-S need not be J-lossless. Finally, consider on H x H, the simple operator s=[S;
I],
(22)
we have es = 11~~1/2-11~11~~112+1152112~
(23)
~13~2~H7
and &J--S= 11~~112-11~~~5~112-11~2112,
x~~x~EH.
(24)
Thus if S,, is lossless on H, S is passive on H x H while J -S is J-active on J-HxH. A simple example of an S which is lossless and J-lossless at the same time is when
s= [ “rl s,,l’
(25)
where both S,, and S2a are lossless on H.
IV.
Synthesis
of Doubly
Active
Scattering
Operators
on a Hilbert
Space
We now discuss the problem of synthesizing a given doubly linear actve scattering operator S on a Hilbert space. It is shown that “any doubly active S on H can be realized by cascade loading its J-lossless dilation on H x H in unit resistors”. This result is, clearly the exact analog of the cascade load synthesis procedure for realizing passive scattering operators on H (6). To exhibit the analogy between. the synthesis of passive and active operators, we briefly consider first the cascade load synthesis of passive operators. Let S, for the moment, be a linear bounded scattering operator from H to H. A dilation (7) of S on H x H is defined to be the matrix operator x = [ ;1
Z:: 1)
(26)
where Xx2, X2, and X22 are linear bounded operators from H to H. Cascading C in an operator St( : H -+ H) as shown in Fig. 1, we have for X : y = cx,
y = [;;I,
cz = E;],
q,52,~1,~2~H,
(27)
or
Vol. 294. No. 5, November 18
1972
Y1=
Sx,+~,,x2
(28)
Y2 =
~21x1
(29)
+x22x2
317
N. Levan and for S,: (30)
x2 = S,Y,* Eliminating
x2 and yz in the above, we find (31)
provided, of course, that the inverse on the right-hand side exists. Thus, it follows that the scattering operator S, of the cascade combination “2 - Si’ (if it exists) is s, = s + Xc,, S,(I - Es2 X,)-l c,,. (32) (1) Now if S is passive, it can be realized by requiring Z to be lossless, and by cascade loading C in unit resistors which corresponds to S, = 0. One such lossless C is the unitary dilation (7) of S given by
Z=
S D, (= [I-S*S-J*) [
-D,*
I*
(= - [I-ss*]* S”
(33)
We have shown” that the cascade load synthesis of a passive S is closely related to the “characteristic operator function (C.O.F.)” (8) of the contraction (i.e. passive) S on H. Indeed by choosing I; to be the unitary dilation (33) and S, = AI, where X is a complex scalar, we have from (32) S,(h) = S -D,.
A(1 - AS*)-l D,,
(34)
where for A(1 - AS*)-1 = (X-1 - S*)-l t o exist, A-l has to belong to the resolvent set p(S*) of S*. Now since X is lossless, the passivity of S,(h) depends only on that of S,(h) which is clearly passive for 1h 1-c 1. Furthermore, since S* is also passive, p(S*) contains the set {A: ]A] < l}. Hence, we conclude that S,(X) of (34) exists and is passive for 1X ( < 1. The passive operator S,(h) when restricted to R(D,): the closure of the range of D, and regarded as an operator from R(D,) to R(D,,) is called the “characteristic operator function” 19,(x) (8) of S: B,(h) = [S-D,,h(I--XS”)D,]IR(D,),
(h]
(35)
(2) Consider now the case of doubly active S. It is clear that the dilation Z of Eq. (33) is no more defined in this case since I-S* S and I - SS* axe now negative operators. Instead, we consider the dilation 2 of S defined by f;=
S A, (= [s*S-I]*) C
* N. Levan: Submitted for publication.
318
A,, ( = [ss* - I]f) Sl”
1,
Joumal of
The Franklin
(36)
Institute
Theory and Applications
of J-Lossless Scattering Systems
where A, However,
and A,, are now well defined, due to Proposition 2, Section II. 2 is no more an unitary operator on H x H, but instead, it is a 1 . J-unitary operator on the J-space -Hx H.* We call 2 the -1 [. “J-unitary dilation” or “J-1ossless dilation” of S. 1 . Theorem I. The -unitary dilation 2 of doubly active S is an -1 L I active operator on H x H.
1
We first show that for 2
Proof:
E=IIxlj2-~~y~~2<0,
x=
,
[1
y=Ec=
;:
(2, YEH x H)
= ll~Il12+llx2112-(llY1112+IIY2112)*
NowsinoeCisJ=[:
(37) (38)
_1]-lossless,wehaveforeJofg E., = (JGx)-(JY,Y)
= 0,
where x and y are as defined in (37). Hence 11%112-11~2112-(IIY1112-IIY2112)
or
= 0
IIx1112-/IY1112
Therefore,
e = 2( II Xl /I2 - II Yl
= 11~2112-IIY2112*
112) = 2( IIx2II2 -
I
(39)
II Y2 l12h
Now y1 = Sz, + A,. x2.
Thus,
4~ = ll~1112-II~~~+As~~2112 = ~Il~,II+II~~,+~,~~,II~~II~,II-II~~,+~,~~,Il~~
(40)
The second term in ( ) on the right-hand side is always negative since both S and A,, are active. Similarly, for the expression of E in terms of I(x2 11and IIyz 11. We conclude that E is negative and therefore f; is active and the theorem is proved. Cascade loading 2 in S, = XI, we have from (31) and (36) : y1 = [S + A&(1
- XS*)-l A,,] x1,
therefore S,(X) = S+A4,h(1-XX*)-iAs*,
(41) (42)
where, as before, h-lsp(S*), which in this case contains the unit disc D, of radius r < 11 S* II-1< 1. Thus we obtain Theorem II. Theorem II. The scattering operator S,(A) = S+AsX(I-XX*)-lAs,
AED,,
143)
obtained by cascade loading the J-lossless dilation 2 of an active S in hl is an active scattering operator on H. * N. Levan: Submitted for publication.
Vol. 294, No. 5, November
1972
319
N. Levan Proof: we have for s, E = II~~l12-IIy~l12~ +Y~EH>
(44)
E = ~ll”~ll2+II~2ll2-(IIY~ll2+IIY2ll2)1+(lIY2/l2-Il~2ll2)~
(45)
where x1, x2, y2, y2 are, as in the passive case, related by (due t’o the cascade combination 2 - XI) Yl = S,(X)
(46)
x17
(47) and x2 = hy,.
(48)
Now, the term in { } on the right-hand side of (45) is just the quantity E of 2. Since 2 is J-lossless, we have, by using (39), E = 2( II x2 /I2 - II Y2112) + ( Ii Y2II2-
IIx2l12).
therefore E = IIx2112-llY2112 = (I~I-l)llY2112~ where use has been made of (48). Hence E
for Ih/< 1
and the theorem is proved. We note from the proof of the theorem that S,(X) would be passive for IX ( > 1, as long as it exists, i.e. as long as p(S*) contains some domain outside of the unit disc. It is clear from (43) that S,(O) is just S, we have Theorem III. Theorem
III.
Any
doubly
active
scattering
operator
S on H can be
realized by cascade loading its
-1ossless dilation in unit resistors. [t -11 Note that, by restricting S,(h) on R(A,), we can also define the C.O.F. for a linear doubly active operator S on a Hilbert space. Finally, it is clear that if S is active, then S/m, where 1~ IIS/I
film Slm= [ [I - (S” S/na2)]f =-
-
[I - (ss*/m2)]*
S*lW&
ia [ (m2A*s)’
-(rn2I-SS”)* s*
1
(49)
1*
Cascade loading Eslm in XI, we find S,(h) =
320
~-(m21ss*)t~(l-~~)-1(~21-s*s)~, IhI< 1. Journalof
The Franklin
(50)
Institute
Theory and Applications
of J-Lossless Scattering Systems
Thus S,(O) realizes, in this case, the passive operator S/m. Consequently, S can be realized from S/m by the scheme of Fig. 2 (9). CIRCULATOR
m+l -m_l
1.Q
FIG. 1.
Conclusion
We have presented the concept of J-lossless scattering operators on a Hilbert space. One immediate application of these operators is to the synthesis of doubly active scattering operators. It is shown that a given doubly active operator can always be realized by cascade loading its J-lossless dilation in unit resistors. This, of course, is similar to the usual method for realizing a passive operator. Also, it is shown that any S with 11 S II> 1 can be realized from the passive operator S/m where m is an arbitrary constant such that 1 < jlS I/ cm. This realization involves the use of circulators. References
(1) R. W. Newcomb,
“Linear Multiport Synthesis”, McGraw-Hill, New York, 1966. (2) Sz-Nagy, “Extensions of linear transformations in Hilbert space which extend beyond this space”, Appendix to F. Riesz and Sz-Nagy, “Functional Analysis”, Frederick Ungar, New York, 1960. (3) R. S. Phillips, “Dissipative operators and hyperbolic systems of partial differential equations”, Trans. Amer. Math. SOL, Vol. 90, pp. 193-254, 1959. (4) M. G. Krein, “Introduction to the geometry of indefinite J-spaces and to the theory of operators in those spaces”, Amer. Math. Sot. Transl. (2), Vol. 93, pp. 103-176, 1970. (5) J. P. Ginzburg, “On J-contractive operator functions”, Pokl. Akad. Nauk SSSR, Vol. 117, pp. 171-173 (in Russian), 1957. (6) R. Se&s, “Synthesis of general linear networks”, SIAM J. Appl. Math., Vol. 16, No. 5, pp. 924-930, Sept. 1968. (7) P. R. Halmos, “Normal dilations and extensions of operators”, Summa Brad Math., Vol. 2, pp. 124-135, 1950. (8) Sz-Nagy and C. Fioas, “Anrtlyse Harmonique des Op&&eurs de 1’Espace de Hilbert”, Masson et Cie. and Akademiai Kiado, 1967. (9) N. Levan, “Synthesis operators”.
Vol. 294, No. 5, November
1972
of active scattering operators by its m-l-derived
passive
321