aa r X i v : . [ m a t h . F A ] N ov Operators of the q –oscillator Franciszek Hugon Szafraniec A bstract . We scrutinize the possibility of extending the result of [ ] to the case of q -deformed oscillator for q real; for this we exploit the whole range of the deformationparameter as much as possible. We split the case into two depending on whether a solutionof the commutation relation is bounded or not. Our leitmotif is subnormality .The deformation parameter q is reshaped and this is what makes our approach ef-fective. The newly arrived parameter, the operator C , has two remarkable properties: itseparates in the commutation relation the annihilation and creation operators from the de-formation as well as it q -commutes with those two. This is why introducing the operator C seems to be far-reaching. q -deformations of the quantum harmonic oscillator (the abbreviation the q - oscillator stands here for it) has been arresting attention of many resulting among other things inquantum groups. Besides realizing the ever lasting temptation to generalize matters, itbrings forth new attractive findings. This paper exhibits the spatial side of the story.The q -oscillator algebra, which is the milieu of our considerations, is that generated bythree objects a + , a − and 1 (the latter being a unit in the algebra) satisfying the commutationrelations a − a + − qa + a − =
1; (1)it goes back to the seventies with [ ] as a specimen. The other versions which appear inthe literature are equivalent to that and this is described completely in [ ] where a list offurther references can be found.Looking for ∗ -representations of (1) usually means assuming that a − = a ∗ + , with theasterisk denoting the Hilbert space adjoint. Thus what we start with is a given Hilbert spaceand the commutation relation S ∗ S − qS S ∗ = I , ( O q , op )in it. Of course, q must be perforce real then; this is what assume in the paper.An easy-going consequence is Sample Theorem.
If S is a weighted shift with respect to the basis { e n } ∞ n = andS ∗ S f − qS S ∗ f = f , f ∈ lin { e n } ∞ n = , then S e n = p + q + · · · + q n e n + , n > . Mathematics Subject Classification.
Primary 47B20, 81S05.
Key words and phrases. unbounded subnormal operator, q -oscillator.Supported at its final stage by the MNiSzW grant N201 026 32 / q -deformations are vastly disseminated in Mathematical Physics and we would like to acknowledge herewith pleasure [ ] for bringing them closer to Mathematics ‘If S is a weighted shift’ – this is usually tacitly assumed when dealing with the relation( O q , op ), like in [ ]. It is sometimes made a bit more explicit in stating that a vacuum vector(or a ground state, depending on denomination in Mathematical Physics an author belongsto) of S exists. The point here (as it was in [ ] for q =
1) is to discuss the case. Itturns out that, like in [ ], subnormality plays an important role in the matter (and this,the case q =
12, 7 ] characterization of solutionsto the CCR). Luckily, the above coincides with our belief that subnormality is the missingcounterpart of complex variable in the quantization scheme.
Preliminary essentialsA short guide to subnormality.
Recall that a densely defined operator A is said tobe hyponormal if D ( A ) ⊂ D ( A ∗ ) and k A ∗ f k k A f k , f ∈ D ( A ). A hyponormal operator N is said to be formally normal if k N f k = k N ∗ f k , f ∈ D ( N ). Specifying more, a formallynormal operator N is called normal if D ( N ) = D ( N ∗ ). Finally, a densely defined operator S is called ( formally ) subnormal if there is a Hilbert space K containing H isometricallyand a (formally) normal operator N in K such that S ⊂ N .The following diagram relates these notions.normal = ⇒ formally normal u ⇓ ⇓ hyponormal t subnormal = ⇒ formally subnormalThough the definitions of formal normality and normality look much alike, with a littledi ff erence concerning the domains involved, the operators they define may behave in atotally incomparable manner. However, needless to say, these two notions do not di ff er atall in the case of bounded operators.If A and B are densely defined operators in H and K resp such that H ⊂ K and A ⊂ B then D ( A ) ⊂ D ( B ) ∩ H , D ( B ∗ ) ∩ H ⊂ P D ( B ∗ ) ⊂ D ( A ∗ ) (2)where P stands for the orthogonal projection of K onto H ; moreover, A ∗ Px = PB ∗ x , x ∈ D ( B ∗ ) . (3)If B closable, then so is A and both A ∗ as well as B ∗ are densely defined. The extension B of A is said to be tight if D ( ¯ A ) = D ( ¯ B ) ∩ H and ∗ - tight if D ( B ∗ ) ∩ H = D ( A ∗ ). If D ( B ) ⊂ D ( B ∗ ) (and this happens for formally normal operators as we already know), thetwo chains in (2) glue together as D ( A ) ⊂ D ( B ) ∩ H ⊂ D ( B ∗ ) ∩ H ⊂ P D ( B ∗ ) ⊂ D ( A ∗ ) . (4)As we have already said a densely defined operator having a normal extension is just sub-normal. However, normal extensions may not be uniquely determined in unbounded caseas their minimality becomes a rather fragile matter, see [ ]; even though the inclusions(4) hold for any of them. Moreover, even if all of them turn into equalities none of thenormal extensions may be minimal of cyclic type (this is what ensures uniqueness); this Description of domains of weighted shifts and their adjoint can be found in [ ]. PERATORS OF THE q –OSCILLATOR 3 will become e ff ective when we pass to the case of q >
1. So far we have got an obviousfact.
Proposition 1.
A subnormal operator S has a normal extension which is both tight and ∗ –tight if and only if D ( ¯ S ) = D ( S ∗ ) . (5) If this happens then any normal extension is both tight and ∗ –tight. Because equality (5) is undoubtedly decisive for a solution of the commutation relationof (any of) the oscillators to be a weighted shift, subnormality is properly settled into thiscontext. q -notions. For x an integer and q real, [ x ] q def = (1 − q x )(1 − q ) − if q , x ] def = x .If x is a non–negative integer, [ x ] q = + q · · · + q x − and this is usually referred to as a basic or q –number. A little step further, the q –factorial is like the conventional, [0] q ! def = n ] q ! def = [0] q · · · [ n − q [ n ] q and so is the q –binomial h mn i q def = [ m ] q ![ m − n ] q ![ n ] q ! . Thus, if − q and x ∈ N the basic number [ x ] q is non–negative.For arbitrary complex numbers a and q one can always define ( a ; q ) k as follows( a ; q ) def = , ( a ; q ) k def = (1 − a )(1 − aq )(1 − aq ) · · · (1 − aq k − ) , k = , , , . . . Then for n > n ] q ! = ( q , q ) n (1 − q ) − n . Moreover, there are (at least) two possibledefinitions of q –exponential functions e q ( z ) def = ∞ X k = q ; q ) k z k , z ∈ ω q , E q ( z ) def = ∞ X k = q ( k )( q ; q ) k z k , z ∈ ω q − , q , , where ω q def = { z ; | z | < } if | q | < C otherwise . These two functions are related via e q ( z ) = E q − ( − z ) , z ∈ ω q , q , . The q oscillatorSpatial interpretation of ( O q , op ). The relation ( O q , op ) has nothing but a symbolicmeaning unless someone says something more about it; this is because some of the solu-tions may be unbounded. By reason of this we distinguish two, extreme in a sense, waysof looking at the relation ( O q , op ):The first meaning of ( O q , op ) is S closable, D is dense in H and D ⊂ D ( S ∗ ¯ S ) ∩ D ( ¯ S S ∗ ), S ∗ S f − qS S ∗ f = f , f ∈ D . ( O q , D )The other is h S f , S g i − q h S ∗ f , S ∗ g i = h f , g i , f , g ∈ D ( S ) ∩ D ( S ∗ ) ( O q , w )and, because this is equivalent to k S f k − q k S ∗ f k = k f k , f ∈ D ( S ) ∩ D ( S ∗ ) F.H. SZAFRANIEC it implies for S to be closable, ( O q , w ) in turn is equivalent to h ¯ S f , ¯ S g i − q h S ∗ f , S ∗ g i = h f , g i , f ∈ D ( ¯ S ) ∩ D ( S ∗ ) . The occurring interdependence, which follows, let us play variation on the theme of( O q , op ).1 o ( O q , D ) with D being a core of S = ⇒ ( O q , w ) and D ( ¯ S ) ⊂ D ( S ∗ ) . Indeed, for f ∈ D ( ¯ S ) there is a sequence ( f n ) n ⊂ D such that f n → f and S f n → ¯ S f .Because S ∗ is closed we get from ( O q , D ) that S ∗ f n → S ∗ f and consequently f ∈ D ( S ∗ ) aswell as ( O q , w ).2 o ( O q , D ) with D being a core of S ∗ = ⇒ ( O q , w ) and D ( S ∗ ) ⊂ D ( ¯ S ) . This uses the same argument as that for 1 o .3 o ( O q , w ) = ⇒ ( O q , D ) with D = D ( S ∗ ¯ S ) ∩ D ( ¯ S S ∗ ).This is because D ( S ∗ ¯ S ) ∩ D ( ¯ S S ∗ ) ⊂ D ( ¯ S ) ∩ D ( S ∗ ).4 o ( O q , w ) and D ( ¯ S ) ∩ D ( S ∗ ) a core of S and S ∗ = ⇒ D ( S ∗ ¯ S ) = D ( ¯ S S ∗ ) . Take f ∈ D ( S ∗ ¯ S ). This means f ∈ D ( ¯ S ) and ¯ S f ∈ D ( S ∗ ). Because of this, picking( f n ) n ∈ D ( ¯ S ) ∩ D ( S ∗ ), we get from ( O q , w ) in limit h S ∗ ¯ S f , g i − q h S ∗ f , S ∗ g i = h f , g i (6)for g ∈ D ( ¯ S ) ∩ D ( S ∗ ) and, because g ∈ D ( ¯ S ) ∩ D ( S ∗ ) is a core of S ∗ , we get (6) to hold for g ∈ D ( S ∗ ). Finally, S ∗ f ∈ D ( ¯ S ). The reverse inequality needs the same kind of argument.The above results in5 o ( O q , w ) and D ( ¯ S ) = D ( S ∗ ) = ⇒ ¯ S satisfies ( O q , D ) on D = D ( S ∗ ¯ S ) = D ( ¯ S S ∗ ) .Remark Notice that when q , − S satisfying ( O q , D ) with D = D ( S ∗ ¯ S ) = D ( ¯ S S ∗ )for D to be a core of S ∗ is necessary and su ffi cient R ( S ∗ S ) to be dense in H .The following is a kind of general observation and settles hyponormality (or bound-edness) in the context of ( O q , D ). Proposition 3. ( a ) For q < and for S satisfying ( O q , D ) , S | D is hyponormal if andonly if S is bounded and k S k (1 − q ) − / . ( b ) For q < and for S satisfying ( O q , D ) , S ∗ | D is hyponormal if and only if S is bounded and k S k (1 − q ) − / . P roof . Write ( O q , D ) as(1 − q ) k S f k = q ( k S ∗ f k − k S f k ) + k f k , f ∈ D . and look at this. (cid:3) The selfcommutator.
Assuming
D ⊂ D ( S S ∗ ) ∩ D ( S ∗ S ) we introduce the followingoperator C def = I + ( q − S S ∗ , D ( C ) def = D . (7)This operator turns out to be an important invention in the matter. In particular there aretwo immediate consequences of this definition. The first says if S satisfies ( O q , D ) with D invariant for both S and S ∗ then D is invariant for C as well and CS f = qS C f , qCS ∗ f = S ∗ C f , f ∈ D . (8)The other is that ( O q , D ) takes now the form S ∗ S f − S S ∗ f = C f , f ∈ D , (9)which means that C is just the selfcommutator of S on D .We would like to know the instances when C is a positive operator. PERATORS OF THE q –OSCILLATOR 5 Proposition 4. (a)
For q > , C > always. (b) For q < , C > if and only if S isbounded and k S k (1 − q ) − / . (c) For S satisfying ( O q , D ) , C > if and only if S ishyponormal. P roof . While (a) is apparently trivial (b) comes out immediately from h C f , f i = k f k + ( q − k S ∗ f k , f ∈ D . For (c) write (using ( O q , D )) with f ∈ Dh C f , f i = k f k + ( q − k S ∗ f k = k f k + q k S ∗ f k − k S ∗ f k = k S f k − k S ∗ f k . (cid:3) Example On the other hand, with any unitary U the operator S def = (1 − q ) − / U (10)satisfies ( O q , D ) if q <
1. The operator S is apparently bounded and normal. Consequently(the Spectral Theorem) it may have a bunch of nontrivial reducing subspaces (even not nec-essarily one dimensional) or may be irreducible and this observation ought to be dedicatedto all those who start too fast generating algebras from formal commutation relations. Proposition 6.
For q < the only formally normal operators satisfying ( O q , D ) are thoseof the form (10) . For q > there is no formally normal solution of ( O q , D ) . P roof . Straightforward. (cid:3) Example An ad hoc illustration can be given as follows. Take a separable Hilbert spacewith a basis ( e n ) ∞ n = −∞ and look for a bilateral (or rather two-sided ) weighted shift T definedas T e n = τ n e n + , n ∈ Z . Then, because T ∗ e n = ¯ τ n − e n − , n ∈ Z , for any α ∈ C and N ∈ Z weget | τ n | = α q n + N + (1 − q n + N )(1 − q ) − = α q n + N + [ n + N ] q for all n if q , | τ n | = α + n if q =
1; this is for all n ∈ Z . The only possibility for the right hand sides to be non–negative(and in fact positive) footnote We avoid weights which are not non–negative, for instancecomplex, as they lead to a unitary equivalent version only. is α > (1 − q ) − for 0 q < α = (1 − q ) − for q <
0; the latter corresponds to Example 10. Thus the only bilateralweighted shifts satisfying ( O q , D ), with D = lin { e n ; n ∈ Z } , are those T e n = τ n e n + , n ∈ Z which have the weights τ n def = p (1 − q ) − , q p α q n + N + [ n + N ] q , α > (1 − q ) − , N ∈ Z , 0 q < , q However,
T violates hyponormality (pick up f = e as a sample) if 0 < q <
1. Also C defined by (7) is neither positive nor negative ( h Ce , e i = a > h Ce − , e − i < T is q − –hyponormal in the sense of [ ]. Anyway, T is apparently unbounded if q >
0. The case of q Example Repeating the way of reasoning of Example 7 we get that the only unilateralweighted shifts satisfying ( O q , D ) are those T , defined as T e n = τ n e n + for n ∈ N , whichhave the weights τ n = q [ n + q , − q . This is so because the virtual, in this case, ‘ τ − ′ is 0 ( T ∗ e = − q < not hyponormal , if 0 q < hyponormal and if1 q the are unbounded and hyponormal ; the two latter are even subnormal (cf. Theorem19 and 21 resp.). F.H. SZAFRANIEC
Remark According to Lemma 2.3 of [ ] for 0 < q < q >
1, due to the same Lemma, the orthogonal sum of that fromExample 8 can be taken into account.
An auxiliary lemma of [ ] . We state here a result, [ ] Lemma 2.4, which autho-rizes the examples above. We adapt the notation of [ ] to ours as well as improve a bitthe syntax of the conclusion therein. Lemma 10.
Let < p < and ε ∈ {− , + } . Assume T is a closed densely definedoperator in H . ThenT ∗ T f − p T T ∗ f = ε (1 − p ) f , f ∈ D ( T ∗ T ) = D ( T T ∗ ) (11) if and only if T is unitarily equivalent to an orthogonal sum of operators of the followingtype: · in the case of ε = T I : f n → (1 − p n + ) / f n + in H = L + ∞ n = H n with each H n def = H ; (II) T II : f n → (1 + q n + A ) / f n + in H = L + ∞ n = −∞ H n with each H n def = H and Abeing a selfadjoint operator in H with sp( A ) ⊂ [ p , and either p or not beingan eigenvalue of A; (III) T III a unitary operator; · in the case of ε = − T IV : f n → ( p n − / f n − in H = L + ∞ n = H n with each H n def = H and alwaysf − def = . A couple of remarks seem to be absolutely imperative.
Remark
The conclusion of Lemma 10 is a bit too condensed. Let us provide with somehints to reading it. First of all the way of understanding the meaning of f n ’s appearing in(I), (II) and (IV) should be as follows: take f ∈ H and define f n as a (one sided or twosides, depending on circumstances) sequence having all the coordinates zero except that ofnumber n which is equal to f . Then, with a definition D ( E ) def = lin { f n ; f ∈ E ⊂ H , n ∈ Z or n ∈ N depending on the case } , one has to guess that D ( T I ) = D ( T IV ) = D ( H ) and D ( T II ) = D ( D ( A )). Passing toclosures in (I), (II) and (IV) we check that T I as well as T IV are everywhere definedbounded operators (use 0 < p <
1) while T II is always unbounded (though satisfying D ( T ∗ II T II ) = D ( T II T ∗ II ) ). Remark
To relate (11) to ( O q , D ) set ε = p = √ q and T = p − p S when 0 < q < ε = − p − = √ q and T = p − p p − S ∗ when q > Positive definiteness from ( O q , D ). The following formalism will be needed.
Proposition 13.
If S satisfies ( O q , D ) with D being invariant for both S and S ∗ , thenS ∗ i S j f = ∞ X k = [ k ] q ! " ik q " jk q S j − k C k S ∗ ( i − k ) f , f ∈ D , i , j = , , . . . , (12) In this matter we have implications 4 o and 5 o on p. 4. PERATORS OF THE q –OSCILLATOR 7 If, moreover, C > then p X i , j = h S i f j , S j f i i = ∞ X k = [ k ] q ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X i = " ik q C k / S ∗ ( i − k ) f i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , f , . . . f p ∈ D . (13) All this under convention S l = ( S ∗ ) l = for l < and h ij i q = for j > i. P roof . Formula (12) is in [ , formula (35)]. Formula (13) is an immediate conse-quence of (12). (cid:3) As a direct consequence of Fact A and (13) we get
Corollary 14.
Suppose S satisfies ( O q , D ) with D being invariant for S and S ∗ as well as D is a core of S . If C > , then p X i , j = h S i f j , S j f i i > , f , . . . f p ∈ D . ( PD ) A useful Lemma.Lemma 15.
Let q > . Consider following conditions: (a) S satisfies ( O q , w ) and D ( ¯ S ) = D ( S ∗ ) ; (b) N ( S ∗ ) , { } and for n = , , . . . f ∈ N ( S ∗ ) = ⇒ ¯ S n f ∈ D ( ¯ S ) , ¯ S ( n − f ∈ D ( S ∗ ) & S ∗ ¯ S n − f = ( n −
1) ¯ S n − f ; (14)(c) there is f , such that ¯ S n f ∈ D ( ¯ S ) , n = , , . . . and ¯ S m f ⊥ ¯ S n for m , n.Then (a) = ⇒ (b) = ⇒ (c) . P roof . (a) = ⇒ (b). The polar decomposition for S ∗ is S ∗ = V | S ∗ | where V is a partialisometry with the initial space R ( | S ∗ | ) and the final space R ( S S ∗ ). Suppose N ( S ∗ ) = { } .Then, because N ( V ) = R ( | S ∗ | ) ⊥ = N ( | S ∗ | ) = N ( ¯ S S ∗ ) = N ( S ∗ ), V is unitary. Since¯ S = | S ∗ | V ∗ , from 5 o we get V | S ∗ | V ∗ = q | S ∗ | + I . Consequently, for the spectra we havesp( | S ∗ | ) ⊂ q sp( | S ∗ | ) + ⊂ [0 , + ∞ ) which is an absurd. Thus N ( S ∗ ) , { } .We show (14) by induction. Of course, N ( S ∗ ) ⊂ D ( ¯ S ) = D ( S ∗ ), which establishes(14) for n =
0. Suppose N ( S ∗ ) ⊂ D ( ¯ S n ) and S ∗ ¯ S n − f = ( n −
1) ¯ S n − f . Then, for g ∈D ( ¯ S ) = D ( S ∗ ), h S ∗ ¯ S n − f , S ∗ g i = ( n − h ¯ S n − f , ¯ S ∗ g i . (15)Because already ¯ S ( n − f ∈ D ( ¯ S ) = D ( S ∗∗ ), we have |h S ∗ ¯ S n − f , S ∗ g i| ≤ C k g k . (16)Because ¯ S ( n − ∈ D ( ¯ S ) = D ( S ∗ ), we can use ( O q , w ) so as to get h ¯ S n f , ¯ S g i = h ¯ S ¯ S ( n − f , ¯ S g i = h S ∗ ¯ S ( n − , S ∗ i + h ¯ S ( n − f , g i . This, by (16), implies ¯ S n f ∈ D ( S ∗ ) = D ( ¯ S ) and, consequently, by (15), gives us S ∗ ¯ S n f = n ¯ S n − f which completes the induction argument. Now a straightforward application of(14) gives ¯ S n ( N ( S ∗ )) ⊂ D ( ¯ S ) ∩ D ( S ∗ ) for n = , , . . . .(b) = ⇒ (c). Take any f ∈ N ( S ∗ ) and using (14) and (12) write h S m f , S n f i = h S n ∗ S m f , f i = min { m , n } X k = [ k ] q ! " mk q " nk q h S ( n − k ) C k S ∗ ( m − k ) f , f i = , m > n . (cid:3) F.H. SZAFRANIEC
A matrix formation.
Suppose q > S is a weighted shift with respect to ( e k ) ∞ k = with the weights ( p [ k + q ) ∞ k = . With S def = S , S n def = q n / S , D n def = q [ n ] q diag( q k / ) ∞ k = , n = , . . . (17)the matrix S D S D . . . S D . . .. . . . . . . . . . . . (18)defines an operator N in L ∞ n = H n , H n = H , with domain composed of all those L n = f n for which f n = n ’s. This matrix, for the familiar creation operatorwas set out in [ ].First we need to determine D ( N ∗ ) and relate it to D ( N ). If 0 < q < D n isbounded. In that case Remark 9 in [ ] gives us D ( N ∗ ) = ∞ M n = D ( S ∗ n ) . (19)If q > S n D − n is bounded. According to Proposition 4.5 in [ ] and Corollary8 in[ ] we can deduce (19) as well. In either case, what we get is the adjoint of N canbe taken as a matrix of adjoints (which is rather an exceptional case). Because the sameargument concerning the adjoint of a matrix operator applies now to N ∗ we can assert thatthe closure operation for the operator N goes entrywise as well. Now, due to the fact thatthe apparent norm equality for N and N ∗ holds on D ( N ), we get essential normality of N .Consequently, S is subnormal and ¯ N is its tight and ∗ –tight normal extension. (20) Subnormality in the q -oscillatorThe case of S bounded. The next says a little bit more about boundedness of solu-tions of ( O q , D ). Proposition 16.
Suppose S is bounded and satisfies ( O q , D ) . ( a ) If q < then k S k > (1 − q ) − / . ( b ) If q < then k S k (1 − q ) − / . ( c ) If q > then no such an S exists. P roof . For (a) look at k S f k = k f k + q k S ∗ f k > k f k + q k S k k f k , for (b) do at k S f k = k f k + q k S ∗ f k k f k + q k S k k f k . For (c) write k S f k = k f k + q k S ∗ f k > q k S k k f k which gives 1 > q . The case of q = (cid:3) The case of q < . Here we get at once
Corollary 17.
For q < the only bounded operator S with norm k S k = (1 − q ) − / satisfying ( O q , D ) is that given by (10) . P roof . By Proposition 16 (a) and Proposition 3 (b) S ∗ | D is hyponormal. On the otherhand, by Proposition 4 (b) and (c) S | D is hyponormal too. Proposition 6 makes the conclu-sion. (cid:3) PERATORS OF THE q –OSCILLATOR 9 Pauli matrices, which are neither hyponormal nor cohyponormal , provide an exampleof operators satisfying ( O − , op ) with norm 1 > − / = (1 − q ) − / . Are there boundedoperators satisfying ( O q , op ) with norm not to be equal (1 − q ) − / for arbitrary q < ff erent from − The case of q < . We list two results which hold in this case
Proposition 18.
Suppose S satisfies ( O q , D ) with D dense in H . If q < , then thefollowing facts are equivalent (i) S is bounded and k S k (1 − q ) − / ; (ii) S is bounded; (iii)
S is subnormal; (iv)
S is hyponormal. P roof . Because of conclusion (a) of Proposition 4 the only remaining implication toargue for is (ii) ⇒ (iii). But, in virtue of (13), this comes out from the Halmos-Bramcharacterization [ ] of subnormality of bounded operators. (cid:3) Theorem 19. If q < , then the following facts are equivalent (i) there is an orthonormal basis ( e n ) ∞ n = in H such that S e n = p [ n + q e n + , n = , , . . . ; (ii) S is irreducible , satisfies ( O q , D ) with some D dense in H , is bounded and k S k = (1 − q ) − / ; (iii) S is irreducible, satisfies ( O q , D ) with some D dense in H , is bounded and k S k (1 − q ) − / ; (iv) S is irreducible, satisfies ( O q , D ) with some D dense in H and is bounded; (v) S is irreducible, satisfies with some D dense in H ( O q , D ) and is subnormal; (vi) S is irreducible, satisfies ( O q , D ) with some D dense in H and is hyponormal. P roof . Proposition 18 establishes the equivalence of (ii) up to (vi).Because sup { p [ n + q ; n > } = (1 − q ) − and for S as being a weighted shift k S k = sup { p [ n + q ; n > } , we get (i) ⇒ (ii).Assume (iv). Because D ( ¯ S ) = D ( S ∗ ), condition (c) of Lemma 15 let calculate theweights of ¯ S while starting with e ∈ N ( N ∗ ). Because S is irreducible the sequence( e n ) ∞ n = is complete. This establishes (i). (cid:3) Remark
From Theorem 19 and Example 5 we get that there are two, of di ff erent nature,solutions of ( O q , D ). Is there any other at all? The case of q > . No bounded solution exits at all, cf. Proposition 16 part (c).Let us memorize what is known already in the bounded case by the following tableau. An operator A is said to be cohyponormal if A ∗ is hyponormal; for unbounded A this may not be the sameas A ∗ | D ( A ) to be hyponormal. Let us recall relevant definitions: a subspace
D ⊂ D ( A ) is invariant for A if A D ⊂ D ; A | D stands for therestriction of A to D . On the other hand, a closed subspace L is invariant for A if A ( L ∩ D ( A )) ⊂ D ( A ); thenthe restriction A ↾ L def = A | L∩D ( A ) . A step further, a closed subspace L reduces an operator A if both L and L ⊥ areinvariant for A as well as P D ( A ) ⊂ D ( A ), where P is the orthogonal projection of ˜ H onto L ; all this is the sameas to require PA ⊂ AP . Then the restriction A ↾ L is called a part of A in L . A is irreducible if it has no nontrivialreducing subspace. Comparing to the more familiar case of bounded operators some nuances become requisitehere. Therefore, if L reduces A , then ( A ↾ L ) = ¯ A ↾ L and ( A ↾ L ) ∗ = A ∗ ↾ L q < q < q normal general SOME Exa. 10
SOME
Exa. 10 unilat. shift SOME
Th. 19 subnormal bilat. shift NONE
Exa. 7
NONE
Exa. 7 others SOME
Exa. 5
SOME
Exa. 5
NONE
Prop. 16(a) unilat. shifts SOME
Th. 19 hyponormal bilat. shift NONE
Exa. 7
NONE
Exa. 7 other SOME
Exa. 5
SOME
Exa. 5
The case of S unbounded. The case of q < . There is no hope to look for subnormal solutions of ( O q , op ) amongweighted shifts, neither one- nor two-sided.The only one-sided weighted shifts satisfying ( O q , op ) are for − < q < O q , op ) are those of Example 7. Theyare normal bilateral weighted shifts. So if there are subnormal operators satisfying ( O q , op )they must not be weighted shifts or bounded operators of norm less or equal (1 − q ) − / , cf.Corollary 17. The case of q < . Lemma 10 does not leave any hope subnormal solutionsdi ff erent than those in Theorem 19 but they must necessarily be bounded. The case of q > . This is the right case for unbounded solutions to exist.
Theorem 21.
For a densely defined closable operator S in a complex Hilbert space H consider the following conditions (i) H is separable and there is an orthonormal basis in it of the form { e n } ∞ n = con-tained in D ( ¯ S ) and such that ¯ S e n = q [ n + q e n + , n = , , . . . ; (21)(ii) S is irreducible, satisfies ( O q , D ) with some D being invariant for S and S ∗ andbeing a core of S , and S is a subnormal operator having a tight and ∗ -tightnormal extension; (iii) S is irreducible, satisfies ( O q , D ) with some D being a core of both S and S ∗ ; (iv) S is irreducible, satisfies ( O q , w ) and D ( ¯ S ) = D ( S ∗ ) ; (v) S is irreducible, satisfies ( O q , w ) with D ( ¯ S ) ∩ D ( S ∗ ) being dense in H , N ( S ∗ ) , { } and ¯ S n ( N ( S ∗ )) ⊂ D ( ¯ S ) ∩ D ( S ∗ ) for n = , , . . . .Then (i) = ⇒ (ii) = ⇒ (iii) = ⇒ (iv) = ⇒ (v) = ⇒ (i) . P roof . The implication (i) = ⇒ (ii) comes out from (20). Proposition 1 leads us from(ii) to (iii), from there using Lemma 15 comes it up to (v). Now, like in the proof ofTheorem 19, calculating the weights rounds up the chain of implications. (cid:3) PERATORS OF THE q –OSCILLATOR 11 Now we visualize this section findings in the following tableau. q < q < q normal general NONE Prop. 6 unilat. shiftsubnormal bilat. shift NONE
Exa. 7 others NONE
Prop. 3(b)
NONE
Prop. 3(a) unilat. shiftshyponormal bilat. shift NONE
Prop. 3(b) others MAY
Prop. 4(a)&(b)
The q oscillator: models in RKHSA general look at. A reproducing kernel Hilbert space H and its kernel K which suitsour considerations is of the form K ( z , w ) def = + ∞ X n = c n z n w n , z , w ∈ D , D = C or D = { z ; | z | < R } . (22)Notice ( √ c n Z n ) + ∞ n = is an orthonormal basis of H .The following fact comes out, as a byproduct, from some general results on subnor-mality in [ ]; we give here an ad hoc argument. Let us make a shorthand notation H ⊂ L ( C , µ ) isometrically. (23) Proposition 22.
There is a measure µ such that (23) holds if and only if there is a Stieltjesmoment sequence ( a n ) + ∞ n = such thata n = c − n , n = , , . . . (24) If this happens than a measure µ can be chosen to be rotationally invariant , that is suchthat µ (e i t σ ) = µ ( σ ) for all t’s and σ ’s. P roof . Suppose (23) to hold. Because ( √ c n Z n ) + ∞ n = is an orthonormal sequence in L ( C , µ ), we have c − n = Z C | z | n µ (d z ) , n = , , . . . Let m µ be the measure on [0 , + ∞ ) transported from µ via the mapping C ∋ z → | z | ∈ [0 , + ∞ ). Then a n def = Z + ∞ r n m µ (d r ) = Z C | z | n µ (d z ) , n = , , . . . (25)satisfies (24) as well as the sequence ( a n ) + ∞ n = is a Stieltjes moment sequence. Or radial as some authors say. If ( a n ) + ∞ n = is any Stieltjes moment sequence with a representing measure m and satis-fying (24) then the rotationally invariant measure µ ( σ ) def = (2 π ) − Z π Z + ∞ χ σ ( r e i t ) m (d r ) d t , σ Borel subset of C (26)makes the imbedding (23) happen. (cid:3) Theorem 23.
Under the circumstances of
Proposition 22 there exists a not rotationallyinvariant measure µ such that (23) holds if and only if there is a sequence ( a n ) + ∞ n = satisfying (24) which is not Stieltjes determinate. P roof . Suppose (23) with µ not rotationally invariant and define ( a n ) + ∞ n = as in (25).Thus there is and s ∈ R such that µ ( τ ) , µ (e i s τ ) for some subset τ of C ; make τ maximalclosed with respect to this property. Let ν be a measure on C transported from µ via therotation z → e − i s z and let m ν be the the measure on [0 , + ∞ ) constructed from ν in the way m µ was from µ , cf. (25). Because, what is a matter of straightforward calculation, m µ and m ν di ff er on {| z | ; z ∈ τ } , we get indeterminacy of ( a n ) + ∞ n = at once.The other way around, if m and m are two di ff erent measures on [0 , + ∞ ) representingthe Stieltjes moment sequence ( a n ) + ∞ n = satisfying (24), then the measure µ on C defined by µ ( σ ) def = (2 π ) − ( s Z a d t Z + ∞ χ σ ( r e i t ) m (d r ) + (1 − s ) Z π a d t Z + ∞ χ σ ( r e i t )( sm (d r ) ,σ Borel subset of C , < s < , < a < π is not rotationally invariant while still (23) is maintained. (cid:3) R´esum´e.
Define two linear operators M and D q acting on functions( M f )( z ) def = z f ( z ) , ( D q f )( z ) def = f ( z ) − f ( qz ) z − qz if q , f ′ ( z ) if q = . (27)It turns out that for a + = M and a − = D q the commutation relation (1) is always satisfied.What Bargmann did in [ ] was to find, for q =
1, a Hilbert space of entire functions suchthat M and D are formally adjoint. This for arbitrary q > H q of analytic functions with the kernel K ( z , w ) def = e q ((1 − q ) z ¯ w ) z , w ∈ | − q | − / ω q where ω q = { z ; | z | < } if 0 < q < C if q > h Z m , Z n i H q = δ m , n [ m ] q !and the operator S = M act as a weighted shift with the weights ( p [ n + q ) as in SampleTheorem on p. 1.Our keynote, subnormality of M now means precisely (23) with some µ is retained.Here we have three qualitatively di ff erent situations:(a) for 0 < q < M is bounded and subnormal, thisimplies uniqueness of µ ;(b) for q = ] and consequently µ is uniquelydetermined as well; PERATORS OF THE q –OSCILLATOR 13 (c) for q > ] though it does plenty ofthose of spectral type in the sense of [ ], which are not unitary equivalent ;explicit example of such, based on [ ], can be found in [ ] (one has to replace q by q − there to get the commutation relation (1) satisfied), an explicit example ofnon radially invariant measure µ is struck out in [ ] and it also comes out fromTheorem 23. The author’s afterword.
The fundamentals of this paper have been presented onseveral occasions for the last couple of years, recently at the Be¸dlewo 9th Workshop
Non-commutative Harmonic Analysis with Applications to Probability . It was Marek Bo˙zejko’scontagious enthusiasm what catalysed converting at long last my distracted notes into acohesive exposition.
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