A caricature of dilation theory
aa r X i v : . [ m a t h . OA ] A p r A CARICATURE OF DILATION THEORY
B.V. RAJARAMA BHAT, SANDIPAN DE, AND NARAYAN RAKSHIT
Abstract.
We present a set-theoretic version of some basic dilation results of operator theory.The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation,inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point outsome natural generalizations and variations.
The basic aim of dilation theory of Hilbert space operators is to realize operators which are apriori not very tractable as compressions of better-behaved operators such as isometries or unitaries.This is a very well developed subject with a number of applications (See [18], [13]) and is also amotivation for studying dilation of quantum dynamical semigroups (See [1], [4]). Here we areobtaining several dilation theory results in a much weaker framework with very little structure.We assume that the reader is familiar with basics of dilation theory of operators. So we do notdwell on explaining usefulness of these ideas nor do we go into nitty gritty of this subject. Thebasic reference for the subject is the classic book “Harmonic analysis of operators on Hilbert space”[18]. A state of the art exposition provided by Orr Shalit [16] is also a good place to begin. Wequote the results we are mimicking without giving proofs.In our setting Hilbert spaces are replaced by sets and bounded operators by arbitrary functions.Injective maps are analogues of isometries and bijective maps are analogues of unitaries. Directsums of Hilbert spaces would be replaced by disjoint unions of sets. In the first section, we beginwith an analogue of Wold decomposition of isometries. We study orbits of injective maps. Likein Operator theory, unilateral shift is the basic model and this is reflected throughout the article.We look at Halmos dilation of contractions and observe that dilations have three ingredients,namely an embedding, an operator in the bigger space and then a compression. This motivatesour constructions. We have simple analogues of Halmos dilation and Sz. Nagy dilation. We havea notion of defect space. Co-invariant minimal dilations would be parametrized be these defectspaces. Going further, we have analogues of intertwining lifting theorem and Sarason’s lemma.In the second section, we look at multivariable theory. We have versions of commuting dilations(such as Ando dilation [2]) and non-commuting dilations (such as Bunce[7], Frazho[9], Popescu[14] dilations). In Section 3, we have an interesting analogue of Berger-Coburn-Lebow theorem oncommuting isometries. In the last Section we describe the possibility of extending our results tosemigroups such as R n or even to general monoids. We observe that it is also possible to have adilation theory where Hilbert spaces are replaced by vector spaces, bounded operators by linearmaps and isometries by injective linear maps.We are calling our presentation a caricature as some features of dilation theory are accentuatedwhereas some other aspects are totally ignored. For instance, we do talk about minimality and soon. Essentially most algebraic structures have been retained whereas analytical concepts such asnorm estimates and inequalities have been filtered out. Naturally, then we do not seem to haveanalogues of results like von Neumann inequality. We do not think of this as a drawback. It ismore interesting than to have all aspects translated in a bijective way.We denote natural numbers { , , , . . . } by N and non-negative integers { , , , . . . } by Z + .Similarly R + = [0 , ∞ ). We would use the notation A ⊔ A to indicate disjoint union and for A ⊆ A , we write A c to indicate the complement A \ A , when the larger set A under considerationis clear. Subject. primary: 47A20;secondary: 04A05.
Key words and phrases. dilation, isometries, injective maps, operator theory. Single variable dilation theory
The first result we wish to consider is the familiar Wold Decomposition of isometries. Let H bea Hilbert space and let V : H → H be an isometry. Take H = T ∞ n =0 V n ( H ) and H = ( H ) ⊥ . Then H , H reduce V , so that V = V ⊕ V , where V = V | H is a unitary and V = V | H is a shift, that is, T ∞ n =0 V n ( H ) = { } . Recall that a subspace W is called a wandering subspace for anisometry V , if V m ( W ) T V n ( W ) = { } for m, n ∈ Z + with m = n. Theorem 1.1. (Wold Decomposition) ( [18] ) Let H be a Hilbert space and let V : H → H be anisometry. Then H decomposes uniquely as H = H ⊕ H where H and H reduce V , V | H is aunitary and T ∞ n =0 V n ( H ) = { } . Definition 1.2.
Let A be a set and let v : A → A be an injective function. Then a subset W of A is said to be wandering for v if v m ( W ) \ v n ( W ) = ∅ for m = n. An injective function v : A → A is said to be a shift if T ∞ n =0 v n ( A ) = ∅ . Theorem 1.3.
Let A be a set and let v : A → A be an injective function. Then A decomposesuniquely as A = A ⊔ A where A and A are left invariant by v and v | A is a shift and v | A is abijection.Proof. Take W = A \ v ( A ). If a ∈ v m ( W ) T v m + k ( W ) for some m, k ∈ Z + , we get a = v m ( w ) = v m + k ( w ) for some w , w ∈ W. As v m is injective, we get w = v k ( w ) . But this is not possibleunless k = 0 as w ∈ A \ v ( A ) . This shows that W is a wandering subset for v . Now take A = ⊔ ∞ n =0 v n ( W ) and A = A c . Then A = A ⊔ A . Clearly v | A is a shift and v | A is a bijection.Suppose that A = A ′ ⊔ A ′ such that v leaves A ′ and A ′ invariant and v | A ′ is a shift and v | A ′ is a bijection. Then A ′ = ⊔ ∞ n =0 ( v | A ′ ) n W ′ where W ′ = A ′ \ v ( A ′ ). Thus we have that W = A \ v ( A ) = ( A ⊔ A ) \ ( v ( A ) ⊔ A ) = A \ v ( A )= ( A ′ ⊔ A ′ ) \ ( v ( A ′ ) ⊔ A ′ ) = A ′ \ v ( A ′ ) = W ′ . Consequently, A = ⊔ ∞ n =0 v n ( W ) = ⊔ ∞ n =0 v n ( W ′ ) = A ′ and hence, A = A ′ . (cid:3) From the Wold decomposition, an isometry on a Hilbert space decomposes as a direct sum ofa pure isometry and a unitary. The pure isometry is just the unilateral shift in l ( Z + ) with somemultiplicity, whereas the unitary part is understood using the spectral theory. We can carry out asimilar analysis for injective functions. Suppose v : A → A is an injective function. For a, b in A ,write a ∼ b if either a = v n ( b ) or v n ( a ) = b for some n ∈ Z + . Clearly this defines an equivalencerelation on A . The corresponding equivalence classes are known as orbits of v .We first classify injective maps which have exactly one orbit. In that direction consider thefollowing examples.(i) Cyclic permutations :
For d ∈ N , take Z d = { , , , . . . , d − } with addition modulo d. Define s d : Z d → Z d by s d ( n ) = n + 1 modulo d. (Note: Z = { } and s (0) = 0 . ). Then s d isbijective and has single orbit.(ii) Bilateral translation:
Define s : Z → Z by s ( n ) = n + 1. Then s is bijective and has singleorbit. (We refrain from calling this as a shift as T n s n ( Z ) = ∅ . )(iii) Unilateral translation/shift:
Define s + : Z + → Z + by s + ( n ) = n + 1 . Then s + is a shiftwith single orbit.A little bit of thought shows that any injective map with single orbit has to be in bijectivecorrespondence with exactly one of these examples. If X is any set, we define 1 X × s : X × Z → X × Z by (1 X × s )( x, n ) = ( x, n + 1) . This will be called as the bilateral translation with multiplicity X . In a similar way, we define1 X × s d ( d -cyclic permutation with multiplicity X ) on X × Z d and 1 × s + ( unilateral shift withmultiplicity X ) on X × Z + . CARICATURE OF DILATION THEORY 3
Now suppose v : A → A is injective. A little bit of thought shows that the action of v onany of the orbits is in bijective correspondence with one of the examples above. Decomposing A into equivalence classes we see that ( A, v ) is in bijective correspondence with (
B, w ) where B = ⊔ d ∈ N ( X d × Z d ) ⊔ X × Z ⊔ X + × Z + , with suitable multiplicity spaces { X d } d ∈ N , X, X Z + (some of these terms could be absent) and w is equal to 1 × s d or 1 × s or 1 × s + in respectivespaces. Cardinalities of these multiplicity spaces are uniquely determined.The most basic dilation theorem for contraction operators on Hilbert spaces is the following. Theorem 1.4. (Halmos Dilation [10] , [16] ) Let T be a contraction ( k T k ≤ on a Hilbert space H . Then U : H ⊕ H → H ⊕ H defined by U = (cid:20) T D T ∗ D T − T ∗ (cid:21) where D T = ( I − T ∗ T ) and D T ∗ = ( I − T T ∗ ) , is a unitary. In other words, every contraction can be enlarged or dilated to a unitary on a larger space. Ifwe identify H with H ⊕
H ⊕ H , the action of U on H restricted to H is same as the action by T , that is, T = P H U | H , We observe that this dilation has three components to it. First, there is the embedding of theoriginal space in a larger space. Then there is a ‘good’ map in the larger space, which is usuallycalled as the dilation. This is followed by a compression (or projection) to the range of the originalspace in the larger space. The last map is usually an idempotent. In probability theory contextsthis map is a conditional expectation map. This is true of most dilation theory results.Here is a simple Halmos type dilation for functions. Let h : A → A be a function. Now wewant a bijection (analogue of unitary) on two copies of A , that is, on B = A × { , } , whichcould be called as dilation. Indeed define i : A → B by i ( a ) = ( a,
0) and u : B → B by u ( a, m ) = ( a, − m ) , ( a, m ) ∈ B . Then i is injective and u is bijective. Further define p : B → B by p ( a, m ) = (cid:26) ( a,
0) if m = 0( h ( a ) ,
0) if m = 1 . Then p is an idempotent ( p = p ) and the range of p is i ( A ) . The quadruple (
B, i, u, p ) has theproperty i ( h ( a )) = p ( u ( i ( a ))) ∀ a ∈ A. Let us recall the Sz. Nagy dilation (See [18]) which is perhaps the most famous dilation of all.Here we are mentioning the ‘isometric dilation’. The isometry can be further dilated to a unitary.
Theorem 1.5.
Let T be a contraction on a Hilbert space H . Then there exists a Hilbert space K containing H with an isometry V : K → K such that T n = P H V n | H ∀ n ∈ Z + Furthermore, ( K , V ) can be chosen such that K = span { V n h : h ∈ H , n ∈ Z + } , and in such a case the pair ( K , V ) is unique up to unitary equivalence. Now we want to get a Sz. Nagy dilation for functions. Here is the basic definition. Here andelsewhere for any function h , h is taken as the identity function. Definition 1.6.
Let A be a non-empty set and let h : A → A be a function. An injective powerdilation (or simply a dilation) of h , is a quadruple ( B, i, v, p ) where B is a set, i : A → B , v : B → B are injective and p : B → B is an idempotent with p ( B ) = i ( A ). Further, i ( h n ( a )) = pv n ( i ( a )) ∀ a ∈ A, n ∈ Z + . Any such dilation is said to be minimal if B = S ∞ n =0 v n ( i ( A )) Two dilations ( B, i, v, p ) and( B ′ , i ′ , v ′ , p ′ ) are said to be bijectively equivalent if there exists a bijection u : B → B ′ suchthat i ′ = ui, v ′ = uvu − , and p ′ = upu − . B.V. RAJARAMA BHAT, SANDIPAN DE, AND NARAYAN RAKSHIT
Theorem 1.7.
Every function h : A → A admits a minimal injective power dilation.Proof. The construction is simple. Take B = A × Z + ; i ( a ) = ( a, , a ∈ A ; v ( a, m ) = ( a, m + 1) , ( a, m ) ∈ B ; p ( a, m ) = ( h m ( a ) , , ( a, m ) ∈ B. It is easily verified that (
B, i, v, p ) is an injective power dilation of h and it is minimal. (cid:3) We will call the dilation constructed in this theorem as the standard dilation of h . Here there isno uniqueness statement. Although the construction gives technically a minimal dilation, it maynot be optimal, for instance when h is injective there is actually no need to enlarge the space. Tomitigate this to some extent we make the following definition. Definition 1.8.
Let h : A → A be a function. A subset D of A is said to be a defect space for h if h | D c is injective (Here D c = A \ D is the complement of D ). A defect space D is said to be aminimal defect space if there is no proper subset of D which is a defect space for h .The defect space measures as to how far the map is from being injective. In this definition, itis important to note that h need not leave D c invariant. We observe that if a family { D j : j ∈ J } of defect spaces of h , forms a chain under inclusion, that is, for any j, k ∈ J either D j ⊆ D k or D k ⊆ D j , then T j ∈ J D j is also a defect space. Now it follows by Zorn’s lemma that every functionadmits a minimal defect space. A minimal defect space is empty if and only if the function isinjective.Let h : A → A be a function and let D be a defect space of h . Define ( B D , i D , v D , p D ) by taking B D = ( D c × { } ) [ ( D × Z + ); i D ( a ) = ( a, , a ∈ A ; v D ( a, m ) = (cid:26) ( h ( a ) , , ( a, m ) ∈ D c × { } ;( a, m + 1) , ( a, m ) ∈ D × Z + ; p D ( a, m ) = ( h m ( a ) , , ( a, m ) ∈ B D . Then it is seen that ( B D , i D , v D , p D ) is a minimal injective dilation. This is same as the standarddilation when D = A. The following definition is motivated by the fact that every minimal isometric dilation V of acontraction T on a Hilbert space H leaves H ⊥ invariant. It is well-known that such a property isimportant while considering dilations of tuples of operators and also in dilation theory of completelypositive maps. This property is called ‘coinvariance’ in [1], ‘regular’ in [5] and co-increasing in [17].We follow Arveson here, as his terminology looks most appropriate in the present situation. Definition 1.9.
An injective dilation (
B, i, v, p ) of a function h : A → A is said to be co-invariant if v ( i ( A ) c ) ⊆ i ( A ) c . It may be noted that the dilation ( B D , i D , v D , p D ) we constructed above using a defect space D is co-invariant. Here is an example of a dilation which is not co-invariant. Example 1.10.
Let A = { } and consider the function h = Id A . Let B = { , } and define v : B → B by v (1) = 2 , v (2) = 1. Let i : A → B be the embedding given by i (1) = 2. Define p : B → B by p (1) = p (2) = 2. Then clearly p is an idempotent with p ( B ) = { } = i ( A ). Furthernote that pv n i (1) = 2 = i ( h n (1)) for all n ∈ Z + . Thus, ( B, i, v, p ) is an injective dilation for h which,obviously, is minimal. But this dilation is not co-invariant as v ( i ( A ) c ) = v ( { } ) = { } = i ( A ). Theorem 1.11.
Let ( B, i, v, p ) be a co-invariant injective dilation of a function h : A → A . Then D = D ( B, i, v, p ) = { a ∈ A : v ( i ( a )) / ∈ i ( A ) } is a defect space for h . Moreover, ( B, i, v, p ) is inbijective correspondence with ( B D , i D , v D , p D ) . CARICATURE OF DILATION THEORY 5
Proof.
Let a, b ∈ D c be such that h ( a ) = h ( b ). Then v ( i ( a )) , v ( i ( b )) ∈ i ( A ) and hence, v ( i ( a )) = pv ( i ( a )) = i ( h ( a )) = i ( h ( b )) = pv ( i ( b )) = v ( i ( b )) and since, i, v are injective, it follows that a = b .Define a map ψ : B D → B by ψ (( a, m )) = v m ( i ( a )) for ( a, m ) ∈ B D . We first show that ψ isa bijection. Let ( a, m ) , ( b, n ) ∈ B D be such that v m ( i ( a )) = v n ( i ( b )). Note that if both a, b ∈ D c ,then we must have that m = n = 0. If one of a, b belongs to D while the other one in D c , say a ∈ D c and b ∈ D , then, of course, m = 0 so that i ( A ) ∋ i ( a ) = ψ (( a, ψ (( b, n )) = v n ( i ( b )). Now if n >
0, the facts that b ∈ D and the dilation is co-invariant together imply that v n ( i ( b )) i ( A )which leads to a contradiction and thus, in this case, we have that m = n = 0. Finally, if a, b both belong to D and if we assume that m = n , say, without loss of generality m > n , then v m ( i ( a )) = v n ( i ( b )) would imply that v m − n ( i ( a )) = i ( b ) ∈ i ( A ). But, as before, it follows fromthe facts a ∈ D , co-invariance of the dilation and m − n > v m − n ( i ( a )) i ( A ) and thus, wearrive at a contradiction. Thus, from ψ (( a, m )) = ψ (( b, n )) we obtain that m = n , from which,using injectivity of v and i , it follows at once that a = b , proving injectivity of the map ψ .We now show that ψ is surjective. Let x ∈ B . If x ∈ i ( A ), say, x = i ( a ) for some a ∈ A , thenclearly x = ψ (( a, x / ∈ i ( A ), we assert that there is ( a, m ) ∈ D × N such that ψ (( a, m )) = x .Since, by minimality of the dilation, B = ∪ ∞ n =0 v n ( i ( A )), x = v n ( i ( a )) for some n > , a ∈ A . If a ∈ D , then x = ψ (( a, n )) and we are done. If a / ∈ D , we must have that n > k be thelargest integer, 1 ≤ k < n , such that v k ( i ( a )) ∈ i ( A ) and v k +1 ( i ( a )) / ∈ i ( A ). Let v k ( i ( a )) = i ( b )where b ∈ A . Clearly, b ∈ D and ψ (( b, n − k )) = v n − k ( i ( b )) = v n ( i ( a )) = x . Therefore, ψ issurjective and hence, bijective. It is easy to verify that vψ = ψv D , pψ = ψp D , and ψi D = i, and we leave these verifications to the reader. Thus ( B, i, v, p ) is in bijective correspondence with( B D , i D , v D , p D ). (cid:3) A contraction T on a Hilbert space H is said to be pure if ( T ∗ ) n converges in strong operatortopology to 0. It is well-known that minimal isometric dilation of a pure contraction is a shift.We wish to have a similar theorem in our set up. But we do not seem to have an exact analogueof pureness. An isometry V is pure if and only if it is a shift that is T ∞ n =0 V n ( H ) = { } . Forcontractions T we know that T ∞ n =0 T n ( H ) = { } , then T is pure, but the converse is not true. Forinstance if T = tI is a scalar with 0 < t <
1, then T is pure but T ∞ n =0 T n ( H ) = H . Theorem 1.12.
Let h : A → A be a function and let ( B, i, v, p ) be a co-invariant, minimal injectivedilation of h . Let D = { a ∈ A : v ( i ( a )) / ∈ i ( A ) } be the associated defect space. Then v is a shiftif and only if T ∞ n =0 h n ( D c ) = ∅ . If h is a function such that T ∞ n =0 h n ( A ) = ∅ , then for every (notnecessarily co-invariant) minimal injective dilation ( B, i, v, p ) of h , v is a shift.Proof. By the previous theorem, (
B, i, v, p ) is in bijective correspondence with ( B D , i D , v D , p D ) . Now if a ∈ T ∞ n =0 h n ( D c ), as v ( i ( a )) = i ( h ( a )) for a ∈ D c , i ( a ) ∈ T ∞ n =0 v n i ( D c ) ⊆ T ∞ n =0 v n ( B )Therefore v is not a shift. Conversely suppose x ∈ T ∞ n =0 v n ( B ) or equivalently x ∈ T ∞ n =0 v nD ( B D ) . Recall that B D = ( D c × { } ) S ( D × Z + ) . If x = ( a,
0) with a ∈ D c , we see that a ∈ T ∞ n =0 h n ( D c )and we are done. Now if x = ( a, m ) for some a ∈ D and m ∈ Z + . For n ≥
1, as x ∈ v n + m ( B ),from looking at the action of v D , we see that there exist a j ∈ D c for n ≥ j ≥
1, such that v D ( a n ,
0) = ( a n − , , v D ( a n − ,
0) = ( a n − , , . . . , v ( a ,
0) = ( a , v D ( a ,
0) = ( a, , v D ( a,
0) = ( a, , v D ( a,
1) = ( a, , . . . , v D ( a, m −
1) = ( a, m ) . Further as h is injective on D c , a j ∈ D c for n ≥ j ≥ h ( a n ) = a n − , h ( a n − ) = a n − , . . . h ( a ) = a , h ( a ) = a. In particular a = h n − ( a n ). So we get a ∈ h n − ( D c ) . As this holds for every n , we get T ∞ n =0 h n ( D c ) = ∅ . This proves the first part.Now for the second part, assume T ∞ n =0 h n ( A ) = ∅ . Let x ∈ T ∞ n =0 v n ( B ) . We know that p ( x ) ∈ i ( A ) . As i is injective there exists unique a ∈ A such that p ( x ) = i ( a ) . Now for any n ∈ Z + , as x ∈ v n ( B ), there exists some b n ∈ B such that x = v n ( b n ). By minimality of the dilation, b n has the form b n = v k ( i ( a n )) for some k ∈ Z + and a n ∈ A. So x = v n + k ( i ( a n )). Then by thedilation property, i ( a ) = p ( x ) = i ( h n + k ( a n )) . Therefore for every n there exists k ≥ a ∈ h n + k ( A ) . Hence a ∈ T ∞ n =0 h n ( A ) . This contradicts purity of h . (cid:3) B.V. RAJARAMA BHAT, SANDIPAN DE, AND NARAYAN RAKSHIT
Here is an example of a dilation which is not co-invariant to illustrate the second part of theprevious theorem.
Example 1.13.
Let A = Z + and let h : A → A be defined by h ( n ) = n + 1. Clearly h is pure.Take B = Z + × { , } . Define i ( a ) = ( a, v ( a,
0) = ( a, , v ( a,
1) = ( a + 2 ,
0) and p ( a,
0) =( a, , p ( a,
1) = ( a + 1 , . It is clear that (
B, i, v, p ) is a minimal injective dilation of h which is notco-invariant. Nevertheless v is a shift with multiplicity 2.The intertwining lifting theorem of Sz. Nagy and C. Foias is well-known. The commutant liftingtheorem is a special case of this. This result is known to have several applications in interpolationtheory and control theory ([8],[13]). Theorem 1.14. (Intertwining lifting theorem [12] ) Let T , T be contractions on Hilbert spaces H , H respectively. Let V , V acting on K , K be minimal isometric dilations of T , T respec-tively. Suppose S : H → H is a bounded operator such that T S = ST . Then there ex-ists a bounded operator R : K → K such that V R = RV , P H R | H ⊥ = 0 , P H R | H = S and k R k = k S k . Conversely if R : K → K is a bounded operator such that V R = RV , and P H R | H ⊥ = 0 , then S : H → H defined by S = P H R | H satisfies T S = ST . Here is an analogue of this theorem.
Theorem 1.15.
Let h : A → A and h : A → A be two functions. Let ( B j , i j , v j , p j ) bestandard dilation of h j for j = 1 , . Suppose s : A → A is a function such that sh = h s . Thenthere exists a function r : B → B such that rv = v r, rp = p r and ri = i s . Conversely if r : B → B is a function satisfying rv = v r and rp = p r, then there exists unique function s : A → A satisfying ri = i s and sh = h s. Proof.
Define r : B → B by r ( a, m ) = ( s ( a ) , m ) for ( a, m ) ∈ B . Recall v j ( a, m ) = ( a, m + 1)for ( a, m ) ∈ B j and i j ( a ) = ( a, j = 1 ,
2. Clearly rv ( a, m ) = r ( a, m + 1) = ( s ( a ) , m +1) = v r ( a, m ) . Similarly rp ( a, m ) = r ( h m ( a ) ,
0) = ( sh m ( a ) ,
0) = ( h m s ( a ) ,
0) = p (( s ( a ) , m ) = p r ( a, m ) and ri ( a ) = r ( a,
0) = ( s ( a ) ,
0) = i s ( a ) . For the converse, consider any a ∈ A . As rp = p r , rp ( a, ∈ i ( A ) . As i is injectivethere exists unique s ( a ) ∈ A such that rp ( a,
0) = ( s ( a ) , s : A → A satisfying ri = i s. Now rp v = p rv = p v r and rp v ( a,
0) = rp ( a,
1) = r ( h ( a ) ,
0) = ( sh ( a ) ,
0) and p v r ( a,
0) = p v ( s ( a ) ,
0) = p ( s ( a ) ,
1) = ( h ( s ( a )) , sh = h s. (cid:3) We can generalize this intertwining lifting for dilations with defect spaces as follows.
Theorem 1.16.
For each j = 1 , , let h j : A j → A j be a function and let D j be a defectspace for h j and consider the injective minimal dilation ( B D j , i D j , v D j , p D j ) of h j . Suppose that s : A → A be a function such that s ( D c ) ⊆ D c , s ( D ) ⊆ D , and sh = h s . Then there existsa function r : B D → B D such that rv D = v D r, rp D = p D r and ri D = i D s . Conversely,if r : B D → B D is a function satisfying rv D = v D r, rp D = p D r , then there exists a uniquefunction s : A → A satisfying ri D = i D s and sh = h s . Moreover, s ( D c ) ⊆ D c .Proof. Similar to the proof of previous Theorem. (cid:3)
Here is a special case of the famous Sarason’s Lemma (See [15], [16]).
Theorem 1.17. (Sarason’s Lemma [15] ) Let H be a closed subspace of a Hilbert space K . Let V : K → K be a bounded operator. Define T : H → H by T = P H V | H . Then V is a power dilation of T , that is, T n = P H V n | H for all n ∈ Z + if and only if there existtwo closed subspaces M ⊆ N ⊆ K , invariant under V such that H = N ⊖ M . Now we present a similar result for injective maps.
CARICATURE OF DILATION THEORY 7
Theorem 1.18.
Let v : B → B be an injective function and let A ⊂ B . Suppose h : A → A isa function such that h ( a ) = v ( a ) for a ∈ A with v ( a ) ∈ A. Suppose A = A \ A where A , A areinvariant under v . Then there exists p : B → B such that p = p , p ( B ) = A and pv n ( a ) = h n ( a ) ∀ n ∈ Z + , a ∈ A. (1) Proof.
All we need to show is that p defined by (1) on S n ∈ Z + v n ( A ) is well-defined. Then it canbe arbitrarily extended to some idempotent on B . Suppose v n ( a ) = v n + m ( a ) for some a , a in A and n, m ∈ Z + . Then as v is injective, a = v m ( a ) . Note that A T A = ∅ . As A isleft invariant under v , if v k ( a ) ∈ A for some k , 1 ≤ k ≤ m , then we get a contradiction as a ∈ A . Consequently v k ( a ) ∈ A , and so v k ( a ) = h k ( a ) for 1 ≤ k ≤ m. Hence a = h m ( a ) and h n ( a ) = h n + m ( a ) . (cid:3) The converse of this theorem is false. Here is a simple example to show this.
Example 1.19.
Let B = Z + , the set of non-negative integers, and let v : B → B be the injectivefunction defined by v (0) = 1 , v (1) = 0 and v ( n ) = n, n ≥
2. Let A = N and let h : A → A be theidentity function. Clearly, h ( n ) = v ( n ) for n ∈ A with v ( n ) ∈ A . Further, if we define p : B → B by p (0) = 1 and p ( n ) = n for n ∈ N , then clearly p is an idempotent with p ( B ) = A and obviously, pv m ( n ) = h m ( n ) for every m ≥ n ∈ A . Suppose A = A \ A for some subsets A , A of B .Note that there are only two choices, namely, A = B, A = { } and A = A, A = ∅ . But clearlyneither A nor { } is v -invariant.We know that Sz. Nagy dilation allows dilation of contractions to unitaries and not just isome-tries. In this context, we have the following result. Theorem 1.20.
Let h : A → A be a function. Then there exists ( C, i, u, p ) where C is a set, i : A → C is injective, u : C → C is a bijection, p : C → C is an idempotent with p ( C ) = p ( A ) such that i ( h n ( a )) = pu n ( i ( a )) a ∈ A, n ∈ Z + , and C = S n ∈ Z u n (( i ( A )) . Proof.
Take C = A × Z , i ( a ) = ( a, , u ( a, n ) = ( a, n + 1) , p ( a, n ) = ( h n ( a ) ,
0) for n ∈ Z + and p ( a, n ) = ( a,
0) for n < . (cid:3) This result is not quite satisfactory as the action of p on ( a, n ) with n < Multivariable dilation theory
One would like to dilate commuting contractions on a Hilbert space to commuting isometries.In this context the following theorem has received a lot of attention.
Theorem 2.1. (Ando Dilation( [2] ) Let T , T be contractions on a Hilbert space H satisfying T T = T T . Then there exists a Hilbert space K containing H as a subspace with a pair ofcommuting isometries V , V on K such that T n T n = P H V n V n | H ∀ n , n ∈ Z + . It is well-known that this theorem can not be extended to triples or to general d -tuples with d ≥ . In contrast, we have the following theorem.
Theorem 2.2.
Let J be an index set and let { h j : j ∈ J } be a commuting family of functions ona set A. Then there exists a quadruple ( B, i, ( v j ) j ∈ J , p ) where B is a set, i : A → B is an injectivemap, v j : A → B , j ∈ J is a commuting family of injective maps, p : B → B is an idempotent,such that i ( h j h j . . . h j k ( a )) = p ( v j v j . . . v j k )( i ( a )) , j , j , . . . , j k ∈ J, a ∈ A and B = { v j v j . . . v j k ( i ( a )) : j , j , . . . , j k ∈ J, a ∈ A } . B.V. RAJARAMA BHAT, SANDIPAN DE, AND NARAYAN RAKSHIT
Proof.
Take B = A × Z J + , , where Z J + , denotes the space of functions from J to Z + taking 0 astheir values except for finitely many points. Then using multi-index notation, for α ∈ Z J + , , h α is defined as h α = Π j ∈ J h α ( j ) j . Here the product means composition of powers of h j ’s and sincethey form a commuting family, the order of composition does not matter. Define i : A → B by i ( a ) = ( a,
0) (Here 0 is the zero function). For j ∈ J , let δ j be the function δ j ( k ) = δ jk . Define v j : B → B by v ( a, α ) = ( a, α + δ j ) and p ( a, α ) = ( h α ( a ) , . It is not difficult to verify that (
B, i, ( v j ) j ∈ J , p ) is a minimal commuting dilation of { h j : j ∈ J } . (cid:3) In the previous theorem also we have not bothered about optimality.
Definition 2.3.
A commuting injective dilation (
B, i, ( v j ) j ∈ J , p ) of a commuting family of func-tions { h j : j ∈ J } on some set A is said to be co-invariant if every v j leaves i ( A ) c invariant.To ensure commutativity of the dilation, we are forced to assume in the following theorem that D c is left invariant by all h j , j ∈ J . This was not the case in one variable. Moreover, now we donot have any uniqueness result. Theorem 2.4.
Let { h j : j ∈ J } be a commuting family of functions on a set A . Let D be a defectspace for each of h j , that is, h j | D c is injective for every j and suppose D c is left invariant by every h j , j ∈ J. Define ( B D , i D , ( v j,D ) j ∈ J , p D ) by B D = D c × { } [ D × Z J + , ; i D ( a ) = ( a, , a ∈ A ; v j,D ( a, α ) = (cid:26) ( h j ( a ) , , ( a, α ) ∈ D c × { } ;( a, α + δ j ) , ( a, α ) ∈ D × Z J + , ; p ( a, α ) = ( h α ( a ) , . Then ( B D , i D , ( v j,D ) j ∈ J , p D ) is a co-invariant, minimal, commuting, injective dilation.Proof. Clearly, ( v j,D ) j ∈ J is a family of injective maps on B D . Now we show that this is a com-muting family. Let ( a, ∈ D c × { } . Then, v i,D v j,D (( a, h i h j ( a ) ,
0) = ( h j h i ( a ) ,
0) = v j,D v i,D ( a, a, α ) ∈ D × Z J + , . Then v i,D v j,D (( a, α )) = ( a, α + δ j + δ i ) = v j,D v i,D (( a, α )).Finally, note that if a ∈ D c , then p D v j ,D v j ,D · · · v j n ,D ( i D ( a )) = p D v j ,D v j ,D · · · v j n ,D (( a, p D ( h j h j · · · h j n ( a ) ,
0) = ( h j h j · · · h j n ( a ) ,
0) = i D ( h j h j · · · h j n ( a )) and if a ∈ D , then p D v j ,D · · · v j n ,D ( i D ( a )) = p D v j ,D v j ,D · · · v j n ,D (( a, p D ( a, δ j + δ j + · · · + δ j n ) = ( h j · · · h j n ( a ) ,
0) = i D ( h j h j · · · h j n ( a )). Thus, ( B D , i D , ( v j,D ) j ∈ J , p D ) is a commuting injective dilation of ( h j ) j ∈ J .Clearly, this dilation is minimal. Observe that i ( A ) c = D × Z J + , \ { } and given ( a, α ) ∈ i ( A ) c (so that a ∈ D ), v j,D ( a, α ) = ( a, α + δ j ) ∈ i ( A ) c . Thus, ( B D , i D , ( v j,D ) j ∈ J , p D ) is a co-invariant,minimal, commuting injective dilation of ( h j ) j ∈ J . (cid:3) For non-commuting operator tuples forming a row contraction, the following dilation to isome-tries with orthogonal ranges is well-known.
Theorem 2.5. ( [7] , [9] , [14] and [16] ) Fix d ≥ and let H be a Hilbert space. Let T j : H → H bebounded operators satisfying T T ∗ + T T ∗ + · · · + T d T ∗ d ≤ I. Then there exists a Hilbert space K containing H as a subspace, with isometries V j : K → K satisfying V ∗ i V j = δ ij ,V ∗ j ( H ) ⊆ H for all j , and T j T j . . . T j k = P H V j V j . . . V j k |H for ≤ j , j , . . . , j k ≤ d and k ≥ . Our analogue of this theorem requires the following definition.
CARICATURE OF DILATION THEORY 9
Definition 2.6.
Let J be an index set and let { h j : j ∈ J } be a family of functions on a set A. Aquadruple (
B, i, ( v j ) j ∈ J , p ), is an injective non-commutative dilation of { h j : j ∈ J } if B is a set, i : A → B is an injective function, for every j , v j : B → B is injective such that v j ( B ) T v k ( B ) = ∅ for j = k , p : B → B is an idempotent with p ( B ) = i ( A ) and i ( h j h j . . . h j k ( a )) = p ( v j v j . . . v j k )( i ( a ))for j , j , . . . , j k in J , k ∈ N and a ∈ A. Such a dilation is said to be minimal if B = { v j v j . . . v j k ( i ( a )) : j , j , . . . , j k ∈ J, k ∈ Z + , a ∈ A } . Theorem 2.7.
Every family of functions { h j : j ∈ J } on a set A admits a minimal injectivenon-commuting dilation.Proof. Define Γ J = { ( j , j , . . . , j k ) : j , j , . . . , j k ∈ J, k ∈ N } [ { ω } where ω is the empty tuple or ‘vacuum’. Take B = A × Γ J ; i ( a ) = ( a, ω ) , a ∈ A ; v j ( a, ω ) = ( a, ( j )); v j ( a, ( j , j , . . . , j k )) = ( a, ( j, j , j , . . . , j k )); p ( a, ω ) = ( a, ω ); p ( a, ( j , j , . . . , j k )) = ( h j h j . . . h j k ( a ) , ω ) . Now the verification of claims made above is easy. (cid:3)
Definition 2.8.
Let { h j : j ∈ J } be a family of functions on a set A . Then a subset D of A issaid to be a joint defect space for { h j : j ∈ J } if h j | D c are injective and h j ( D c ) \ h k ( D c ) = ∅ for j = k. The condition in the previous definition can also be stated as H : J × A → A defined by H ( j, a ) = h j ( a ) is injective on J × D c .Now suppose D is a joint defect space for { h j : j ∈ J } . Consider ( B D , i D , ( v j,D ) j ∈ J , p ), where B D = D c × { ω } S D × Γ J , i D ( a ) = ( a, ω ) , v j,D ( a, ω ) = ( h j ( a ) , ω ) if a ∈ D c and v j,D ( a, ω ) =( a, ( j )) if a ∈ D and v j,D ( a, ( j , j , . . . , j k )) = ( a, ( j, j , j , . . . , j k )) . Also p ( a, ω ) = ( a, ω ) and p ( a, ( j , j , . . . , j k )) = ( h j h j . . . h j k ( a ) , ω ) . Then ( B D , i D , ( v j,D ) j ∈ J , p ) is a non-commutative injec-tive dilation of { h j : j ∈ J } . We also observe that if v j,D ( i D ( a )) / ∈ i D ( A ) then v k,D ( i D ( a )) / ∈ i D ( A )for every k ∈ J . This is a property crucial for the next theorem. Theorem 2.9.
Let { h j : j ∈ J } be a family of functions on a set A . Let ( B, i, ( v j ) j ∈ J , p ) be aninjective, non-commutative, minimal, co-invariant dilation of { h j : j ∈ J } and has the propertythat if for some a ∈ A and j ∈ J , v j ( i ( a )) / ∈ i ( A ), then v k ( i ( a )) / ∈ i ( A ) for every k ∈ J . Set D = { a ∈ A : v j ( i ( a )) / ∈ i ( A ) for some j ∈ J } . Then D is a joint defect space for { h j : j ∈ J } and ( B D , i D , ( v j,D ) j ∈ J , p D ) is bijectively isomorphicto ( B, i, v, p ). Proof.
Observe that given any a ∈ A , either v j ( i ( a )) ∈ i ( A ) for every j ∈ J or v j ( i ( a )) / ∈ i ( A )for all j ∈ J . We first show that D is a joint defect space for { h j : j ∈ J } . Let j ∈ J and let a, b ∈ D c be such that h j ( a ) = h j ( b ). As a, b ∈ D c , both v j ( i ( a )) and v j ( i ( b )) belong to i ( A ). Thus, v j ( i ( a )) = pv j ( i ( a )) = i ( h j ( a )) = i ( h j ( b )) = pv j ( i ( b )) = v j ( i ( b )) and hence, a = b . Also for j = k , h j ( D c ) ∩ h k ( D c ) = ∅ for if h j ( D c ) ∩ h k ( D c ) = ∅ , say x ∈ h j ( D c ) ∩ h k ( D c ), then x = h j ( a ) = h k ( b )for some a, b ∈ D c which would imply that v j ( i ( a )) = v k ( i ( b )) and consequently, v j ( B ) ∩ v k ( B ) = ∅ ,a contradiction. Hence, h j ( D c ) ∩ h k ( D c ) = ∅ for j, k ∈ J with j = k .Consider the map ψ : B D → B defined by ψ ( a, ω ) = i ( a ) and ψ ( a, α ) = v α ( i ( a )) for ( a, α ) ∈ B D with α = ω . We show that ψ is injective. Let ( a, α ) , ( b, β ) ∈ B D be such that ψ ( a, α ) = ψ ( b, β ).If one of a, b , say a , belongs to D and the other one, that is, b belongs to D c , then we must have that β = ω and α = ω and hence, v α ( i ( a )) = i ( b ). Let α = ( j , j , · · · , j n ) where n ≥
1. As a ∈ D , v j ( i ( a )) / ∈ i ( A ) for every j ∈ J and so, in particular, v j n ( i ( a )) / ∈ i ( A ). Now the co-invariance ofthe dilation allows us to conclude that v α ( i ( a )) / ∈ i ( A ), leading to a contradiction. Thus, either a, b ∈ D or a, b ∈ D c . If both a and b belong to D c , then clearly, α = β = ω from which it followsthat i ( a ) = i ( b ) and so, a = b . Let a, b ∈ D . If α = β = ω , then obviously a = b . We assert thatit can not happen that one of α, β is ω and the other one is different from ω for if, say β = ω and α = ω , then v α ( i ( a )) = i ( b ) which is a contradiction since similar argument as before establishesthat v α ( i ( a )) / ∈ i ( A ). Let α = ( j , j , · · · , j n ) and β = ( k , k , · · · , k m ) where m, n ≥
1. Then v j · · · v j n ( i ( a )) = v k · · · v k m ( i ( b )). If possible let n > m . It follows from v j ( B ) ∩ v k ( B ) = ∅ for j = k that j = k , · · · , j m = k m and thus, i ( b ) = v j m +1 · · · v j n ( i ( a )), a contradiction. Thus m = n and once again applying the fact that v j ( B ) ∩ v k ( B ) = ∅ for j = k we obtain that i ( a ) = i ( b ) sothat a = b . Thus ψ is injective.Next we show that ψ is surjective. Let x ∈ B . If x ∈ i ( A ), say, x = i ( a ) for some a ∈ A , then ψ ( a, ω ) = i ( a ) = x . Assume that x / ∈ i ( A ). It follows from from minimality of thedilation that x = v j v j · · · v j n ( i ( a )) where n ≥ , a ∈ A and j , · · · , j n ∈ J . Clearly, if a ∈ D ,then ψ (( a, ( j , j , · · · , j n ))) = x . If a ∈ D c , v j n ( i ( a )) ∈ i ( A ) and let k be the smallest positiveinteger, 1 < k ≤ n , such that v j k v j k +1 · · · v j n ( i ( a )) ∈ i ( A ) and v j k − v j k · · · v j n ( i ( a )) / ∈ i ( A ). Let v j k v j k +1 · · · v j n ( i ( a )) = i ( b ) where b ∈ A . As v j k − ( i ( b )) / ∈ i ( A ), b ∈ D and ψ (( b, ( j , · · · , j k − ))) = v j · · · v j k − ( i ( b )) = x . Therefore, ψ is surjective and hence, bijective.One can easily verify that ψv j,D = v j ψ, ψp D = pψ, and ψi D = i, for every j ∈ J and consequently, ( B, i, v, p ) is bijectively isomorphic to ( B D , i D , ( v j,D ) j ∈ J , p D ). (cid:3) Berger, Coburn and Lebow Theorem
A theorem of Berger, Coburn and Lebow ([3]) describes the structure of two commuting isome-tries. We follow the exposition of Maji, Sarkar and Sankar [11]. For a recent non-trivial applicationof this theorem see Bhattacharyya, Kumar and Sau [6]. The result is as follows.Let V , V be two commuting isometries on a Hilbert space H and V = V V . Then by Wolddecomposition of V , H = H ⊕ H decomposing V as V = V | H ⊕ V | H , where V | H is a unitaryand V | H is a shift with some multiplicity. So up to unitary isomorphism H = H ⊗ M , and V | H = M z ⊗ I D , where M z is the standard shift isometry on the Hardy space and M is amultiplicity space.It is not hard to see that H and H reduce V , V and so they decompose say as V = V ⊕ V and V = V ⊕ V . Of course, this may not be Wold decomposition of V , V . However V and V are commuting unitaries. Further, V and V are commuting isometries related by a formulaas below. Theorem 3.1. (BCL Theorem [3] ) Under the set up as above, there exists a projection P on D and a unitary U on D such that V = S ⊗ U ∗ P + I ⊗ U ∗ ( I − P ) and V = I ⊗ P U + S ⊗ ( I − P ) U. Conversely, any pair of a projection P and a unitary U on D would give a commuting pair ofisometries by this formula. Let A be a non-empty set and let v : A → A be an injective map. Suppose v factorizes as v = v v = v v where v i : A → A are injective for i = 1 , . Consider the Wold type decompositionof v . So take A = T ∞ n =0 v n ( A ) and A = A c . Using commutativity of v , v , if a = v n ( a ), then v ( a ) = v n ( a ) where a = v ( a ) . This shows that A is invariant under v . Now if v ( a ) = v n +1 ( a ) then a = v n ( a ) where a = v ( a ) . This shows that if v ( a ) ∈ A , then a ∈ A . So A is also invariant under v . Similarly A and A are also invariant under v . Now v | A is a bijection.So A = v ( A ) = v v ( A ) ⊆ v ( A ) ⊆ A . Hence v , v are surjective on A . Consequently, v , v on A are commuting bijections. So, to understand the structure of v it suffices to consider v | A .Hence, without loss of generality here after we assume that v is a shift.Take W = A \ v ( A ). As observed before W is a wandering subset for v. Similarly take W i = A \ v i ( A ) for i = 1 , . Now if a ∈ W , then either a ∈ W or a = v ( a ) for some a . But then a has to be in W , otherwise we would have a ∈ v v ( A ) . Arguing this way from the commutativityof v , v we get W = W ⊔ v ( W ) = v ( W ) ⊔ W . Define u : W → W , that is, from W ⊔ v ( W ) to v ( W ) ⊔ W , by u ( w ) = v ( w ) and u ( v ( w )) = w for w ∈ W , w ∈ W . It is easily seen that u is a bijection with inverse u − given by u − ( v ( w )) = w and u − ( w ) = v ( w ) . From Wold type decomposition, as v is a shift, A = ⊔ ∞ n =0 v n ( W ) = W ⊔ v ( W ) ⊔ v ( W ) ⊔ · · · . For n ≥ u to v n ( W ) by taking u ( v n ( w )) = v n ( u ( w )) for any w ∈ W. Then u : A → A is a bijection commuting with v . Take C = ⊔ ∞ n =0 v n ( W ) and C = ⊔ ∞ n =0 v n ( v ( W )),so that A = C ⊔ C Moreover, v : A → A is given by v ( x ) = (cid:26) u − ( x ) if x ∈ C u − ( v ( x )) if x ∈ C . Similarly, v : A → A is given by v ( x ) = (cid:26) u ( x ) if x ∈ u − ( C ) u ( v ( x )) if x ∈ u − ( C ) . This can be written using unilateral shift as follows. Take B = W × Z + . Let 1 × s + : B → B bethe canonical unilateral shift with wandering space W (identified with W × g : A → B defined by g ( v n ( w )) = ( w, n ) for w ∈ W and n ∈ Z + is a bijection such that v = g − (1 × s + ) g. Take s i = gv i g − for i = 1 ,
2. Then s ( w, n ) = (cid:26) ( u − ( w ) , n ) ( w, n ) ∈ W × Z + ;( u − ( w ) , n + 1) ( w, n ) ∈ v ( W ) × Z + ;and s ( w, n ) = (cid:26) ( u ( w ) , n ) ( w, n ) ∈ W × Z + ;( u ( w ) , n + 1) ( w, n ) ∈ v ( W ) × Z + . Recall that W decomposes as W = W ⊔ v ( W ), and u : W → W is a bijection such that u ( W ) = v ( W ) = W c . In other words we have the following theorem.
Theorem 3.2.
With notation as above, ( A, v, v , v ) is in bijective correspondence with ( W × Z + , W × s + , s , s ) where, s = ( u − × id. ) | W × Z + + ( u − × s + ) | W c × Z + ; s = ( u × id. ) | u − ( W c ) × Z + + ( u × s ) | u − ( W ) × Z + . Now we extend our BCL type theorem to families of maps. Let { v j : j ∈ { , , . . . n }} be acommuting family of injective maps on a set A . Assume that v = v v . . . v n is a shift. Take W = A \ v ( A ) and W j = A \ v j ( A ) for j = 1 , , . . . , n. Let S n denote the group of permutations of { , , . . . , n } . Now for any σ ∈ S n , W has a decomposition given by W = W σ (1) ⊔ v σ (1) ( W σ (2) ) ⊔ v σ (1) v σ (2) ( W σ (3) ) ⊔ · · · ⊔ v σ (1) v σ (2) · · · v σ ( n − ( W σ ( n ) ) , and we use the notation W ( σ ) to denote this decomposition of W . For τ, σ ∈ S n , let u στ denotethe bijection from W ( σ ) to W ( τ ) defined as follows: u στ ( v σ (1) v σ (2) · · · v σ ( k − ( w )) = v τ (1) v τ (2) . . . v τ ( r − ( w )for w ∈ W σ ( k ) , with r being chosen such that τ ( r ) = σ ( k ). For k = 0 , , · · · , n −
1, let σ k ∈ S n begiven by σ k ( j ) = ( k + j, for 1 ≤ j ≤ n − k,k − n + j, for n − k + 1 ≤ j ≤ n. and we consider the family { u σ k σ k − : 1 ≤ k ≤ n } of bijections of W where the notation σ n standsfor σ . Using these bijections, we can describe the maps v k (1 ≤ k ≤ n ) on W as follows: v k = ( u σ k σ k − , on W \ Q ni =1 ,i = k v i ( W k ) ,vu σ k σ k − , on Q ni =1 ,i = k v i ( W k ) . Now, as before, using the fact that A = ⊔ ∞ m =0 v m ( W ), we may extend the bijections u σ k σ k − of W tobijections of A by setting u σ k σ k − ( v m w ) = v m ( u σ k σ k − w ) for w ∈ W and m ≥
0. Consequently, v k = ( u σ k σ k − , on ⊔ ∞ m =0 v m ( W \ Q ni =1 ,i = j v i ( W k )) ,vu σ k σ k − , on ⊔ ∞ m =0 v m ( Q ni =1 ,i = j v i ( W k )) . Note that as before, g − (1 × s + ) g = v . Set s k = gv k g − for k = 1 , , · · · , n . We then have thefollowing result. Theorem 3.3.
With the same notations as in Theorem 3.2, ( A, v, v , v , · · · , v n ) is in bijectivecorrespondence with ( W × Z + , W × s + , s , s , · · · , s n ) where s k = ( u σ k σ k − × id. ) | W ′′ k × Z + + ( u σ k σ k − × s + ) | W ′ k × Z + , ≤ k ≤ n ; and W ′ k = Q ni =1 ,i = k v i ( W k ) = W \ ( u σ k − σ k ( W ( σ k − ) \ W k )) , W ′′ k = W \ W ′ k . Generalizatons and variations
We considered dilations of { h n : n ∈ Z + } for a map h : A → A. Now instead of Z + we canconsider R + or more general monoids. Dilation theory on general monoids have been consideredby many authors, see for instance [17]. Recall that a monoid is a set S with an associative binaryoperation (say ‘ . ’) and an identity element (say 1). The associative operation need not be abelian.It seems eminently feasible to extend most of what we did in previous sections to this setting. Hereis the standard dilation in this setting. Theorem 4.1.
Let S be a left cancellative monoid. Suppose A is a set and { h s : s ∈ S } is a familyof functions such that h s.t = h s ◦ h t for all s, t ∈ S and h = id. . Then there exists a quadruple ( B, i, { v s : s ∈ S } , p ) where B is a set, i : A → B , v s : B → B are injective functions, v s.t = v s ◦ v t for all s, t in S , v = id., p : B → B is idempotent with p ( B ) = i ( A ) such that p ( v s ( i ( a ))) = i ( h s ( a )) ∀ s ∈ S, a ∈ A and B = S s ∈ S { v s ( i ( a )) : s ∈ S, a ∈ A } . Proof.
Take B = A × S . Define i : A → B by i ( a ) = ( a, v s ( a, t ) = ( a, s.t ) and p ( a, s ) = ( h s ( a ) , . The left cancellative property of the monoid S ensures that v s are isometries. (cid:3) Instead of working with sets and functions, we can try to develop the dilation theory workingwith vector spaces and linear maps. Now bounded operators on Hilbert spaces gets replaced byarbitrary linear maps, isometries by injective linear maps and unitaries by bijective linear maps.Direct sums of Hilbert spaces gets replaced by direct sums of vector spaces. Here is a formaldefinition and a sample result.
Definition 4.2.
Let A be a vector space and let h : A → A be a linear map. A quadruple( B, i, v, p ) is said to be a minimal injective linear dilation of h if B is a vector space, i : A → B isan injective linear map, v : B → B is an injective linear map, p : B → B is an idempotent linearmap with p ( B ) = i ( A ), satisfying i ( h n ( a )) = p ( v n ( i ( a )) ∀ n ∈ Z + , a ∈ A. Such a dilation is said to be minimal if B = span { v n ( i ( a )) : n ∈ Z + , a ∈ A } . Theorem 4.3.
Every linear map h : A → A on a vector space admits a minimal injective lineardilation.Proof. Take B = A Z + , the space of functions from Z + to A which take value 0 at all but finitelymany points. It is a vector space under natural linear operations. Define i : A → B by i ( a )(0) = a and i ( a )( n ) = 0 for n = 0 . Define v : B → B by v ( b )(0) = 0 and v ( b )( n ) = b ( n −
1) for n ≥ . Finally define p : B → B by p ( b ) = P ∞ n =0 ih n ( b ( n )) . This map p is well defined as b ( n ) = 0 for allbut finitely many n . It is easy to see that ( B, i, v, p ) is a minimal injective linear dilation of h. (cid:3) CARICATURE OF DILATION THEORY 13
We observe that the construction of dilation here is similar to the standard dilation of functions(See Theorem 1.7), however there are certain differences. Now the addition operation of vectorspaces plays a non-trivial role.
Acknowledgements.
The first author thanks J C Bose Fellowship of SERB (India) for financialsupport. The other two authors are supported by the NBHM (India) post-doctoral fellowship andthe Indian Statistical Institute. We thank T. Bhattacharyya for some useful discussions on thetopic.
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