Base Subsets of the Hilbert Grassmannian

Innovations in Incidence Geometry ACADEMIA Volume 4 (2006), Pages 63–68 PRESS ISSN 1781-6475

Base subsets of the Hilbert Grassmannian

Mark Pankov

Abstract Let H be a separable Hilbert space. We consider the Hilbert Grassman- nian (H) consisting of closed subspaces having inﬁnite dimension and G∞ codimension and show that every bijective transformation of (H) pre- G∞ serving the class of base subsets is induced by an element of GL(H) or it is the composition of the transformation induced by an element of GL(H) and the bijection sending a subspace to its orthogonal complement.

Keywords: Hilbert Grassmannian, base subset, inﬁnite-dimensional topological projective space MSC 2000: 46C05, 51E24

1 Introduction

We start from the classical Grassmannian (V ) consisting of all k-dimensional Gk subspaces of an n-dimensional vector space V . A base subset of k(V ) is the set n G of k distinct k-dimensional subspaces spanned by vectors of a certain base of V . This construction is closely related with Tits buildings [5]: the Grassmanni- ans (V ) (k = 1, . . . , n 1) are the shadow spaces of the building associated Gk − with the group GL(V ) and their base subsets are the shadows of the correspond- ing apartments. It was proved in [2] that transformations of (V ) preserving Gk the class of base subsets are induced by a semilinear isomorphism of V to itself or to V ∗ (the second possibility can be realized only for the case when n = 2k); a more general result can be found in [3]. Suppose that V is an inﬁnite-dimensional vector space and dim V = . The ℵ group GL(V ) acts on the set of all subspaces of V and the orbits of this action S are called Grassmannians. There are the following three types of Grassmanni- ans: (V ) := S dim S = α, codim S = Gα { ∈ S | ℵ } 64 Pankov

α(V ) := S dim S = , codim S = α G { ∈ S | ℵ } for any cardinality α < and ℵ (V ) := S dim S = codim S = . Gℵ { ∈ S | ℵ } As in the finite-dimensional case, we define base subsets of Grassmannians. It was shown in [4] that every bijective transformation of (V ), α < is Gα ℵ induced by a semilinear automorphism of V . The methods of [4] can not be applied to other Grassmannians; this is related with the fact that V and V ∗ have different dimensions and the duality principles do not hold for infinite- dimensional vector spaces.

Now suppose that V is a topological vector space. Denote by ΠV the projective space associated with V . The topology of V induces a topology on ΠV and we talk about an inﬁnity-dimensional topological projective space. A collineation of ΠV to itself will be called closed if it preserves the class of closed subspaces. It follows from Mackey’s results [1] that every closed collineation of ΠV to itself is induced by an invertible bounded linear operator if V is a real normed space. In the general case, closed collineations are not determined. In the present note we give a geometrical characterization of closed collineations of the projective space over a separable Hilbert space.

2 Result

Let H be a separable Hilbert space (real or complex). We write (H) for the G∞ Hilbert Grassmannian consisting of all closed subspaces with inﬁnite dimension and codimension. Let B = xi i be a base of H (possibly non-orthogonal). { } ∈ N The set of all elements of (H) spanned by subsets of B is called the base G∞ subset of (H) associated with (or deﬁned by) the base B. G∞ Every closed collineation of ΠH to itself induces a bijective transformation of (H) preserving the class of base subsets. The bijection G∞ : (H) (H) ◦ G∞ → G∞ sending subspaces to their orthogonal complements also preserves the class of base subsets.

Theorem 2.1. If f is a bijective transformation of (H) preserving the class of G∞ base subsets, then f is induced by a closed collineation of ΠH to itself or it is the composition of the transformation and the transformation induced by a closed ◦ collineation of ΠH to itself. Base subsets of the Hilbert Grassmannian 65

3 Preliminaries

Let B = xi i be base of H, and let be the associated base subset of (H). { } ∈ N B G∞ We denote by P the 1-dimensional subspace containing x and write and i i B+i i for the sets of all elements of which contain Pi or do not contain Pi, B− B respectively. Then i consists of all elements of contained in the hyperplane B− B S := B x . i \ { i} We say that is an exact subset if there is only one base subset of (H) X ⊂ B G∞ containing ; otherwise, is said to be inexact. X X Remark 3.1. It is trivial that is exact if and only if for each i N the intersec- X ∈ tion of all U containing P coincides with P . For example, ∈ X i i

ij := ( +i +j) i i = j R B ∩ B ∪ B− 6 is an inexact subset. Indeed, the intersection of all elements containing Pk coincides with P if k = i, however for k = i this intersection is P + P . Every k 6 i j element of intersects P + P by P . This means that B \ Rij i j i U Rij ∪ { } is exact and the inexact subset is maximal. Conversely, every maximal inex- Rij act subset of coincides with certain (the proof of this fact is similar to the B Rij proof of Lemma 1 in [4]).

Let 0 be the base subset of (H) deﬁned by a base B0 = xi0 i . We write B G∞ { } ∈ N +0 i and 0 i for the sets of all elements of 0 which contain Pi0 or do not contain B B− B Pi0 (respectively); here Pi0 is the 1-dimensional subspace containing xi0 . We also deﬁne S0 := B x . i 0 \ { i0 } 1 A bijection g : 0 is called special if g and g− map inexact subsets to B → B inexact subsets.

Lemma 3.2. If g : 0 is a special bijection then there exists a bijective B → B transformation δ : N N such that →

g( +i) = +0 δ(i), g( i) = 0 δ(i) i N B B B− B− ∀ ∈ or

g( +i) = 0 δ(i), g( i) = +0 δ(i) i N. B B− B− B ∀ ∈ Proof. This is similar to the proof of Lemma 3 in [4]. 66 Pankov

We say that a special bijection g : 0 is of ﬁrst type if the ﬁrst equality of B → B Lemma 3.2 holds; otherwise, g is said to be of second type.

Lemma 3.3. Let g and δ be as in the previous lemma. Let also S, U . If g is of ∈ B ﬁrst type then

S U = P g(S) g(U) = P 0 and S + U = S g(S) + g(U) = S0 . ∩ i ⇔ ∩ δ(i) i ⇔ δ(i) If g is of second type then

S U = P g(S) + g(U) = S0 and S + U = S g(S) g(U) = P 0 . ∩ i ⇔ δ(i) i ⇔ ∩ δ(i) Proof. The equality S U = P holds if and only if S, U belong to and there ∩ i B+i is no j = i such that contains both S, U. Similarly, we have S + U = S if 6 B+j i and only if S, U belong to i and there is no j = i such that j contains both B− 6 B− S, U. Lemma 3.2 gives the claim.

For every 1-dimensional subspace P we denote by [P ] the set of all elements of (H) containing P . If S is a closed hyperplane then we write [S] for the set G∞ of all elements of (H) contained in S. G∞ Lemma 3.4. For any i N and any U [Pi] there exist M, N +i such ∈ ∈ \ B ∈ B that M N = P , codim M + N > 1, ∩ i and M, N, U are contained in a certain base subset of (H). G∞

Proof. Let Uj j X be a countable collection of elements of such that { } ∈ ⊂ N B U U = 0 if k = n. k ∩ m 6 For every j X we choose P U U (this is possible, since U = U ) and ∈ j ∈ j \ 6 j denote by T the element of spanned by P and all P , j X. Then U T = P . B i j ∈ ∩ i We take any M, N contained in T and such that M N = P . These ∈ B+i ∩ i subspaces are as required.

Since : ( ) is a special bijection of second type, we have the ◦|B B → ◦ B following dual version of Lemma 3.4.

Lemma 3.5. For any i N and any U [Si] there exist M, N i such ∈ ∈ \ B ∈ B− that M + N = S , dim M N > 1, i ∩ and M, N, U are contained in a certain base subset of (H). G∞ Base subsets of the Hilbert Grassmannian 67

4 Proof of the theorem

Let f be a bijective transformation of (H) preserving the class of base sub- G∞ sets. We consider an arbitrary base subset (H). Then 0 := f( ) B ⊂ G∞ B B is a base subset and f : 0 is a special bijection. Suppose that B = |B B → B xi i and B0 = xi0 i are bases associated with and 0, respectively. Let { } ∈ N { } ∈ N B B Pi, P 0, Si, S0 and δ : N N be as in the previous section. It is clear that we can i i → assume that δ is identical. We have to consider the following possibilities:

(A) f is a special bijection of ﬁrst type, |B (B) f is a special bijection of second type. |B

Case (A). First we claim that

f([Pi]) = [P 0] i N. i ∀ ∈ Let U, M, N be as in Lemma 3.4. By Lemma 3.3,

f(M) f(N) = P 0, codim f(M) + f(N) > 1. (1) ∩ i If is a base subset containing U, M, N then (1) together with Lemma 3.3 show B that the restriction of f to is a special bijection of ﬁrst type. Bases associated B withe and its f-image contain vectors lying in Pi and Pi0 (respectively), and B 1 Lemma 3.2 implies that f(Ue ) belongs to [P 0]. Thus f([P ]) [P 0]. Since f − i i ⊂ i preservese the class of base subsets, we can also prove the inverse inclusion. The claim follows. Using Lemma 3.5 we show that

f([Si]) = [S0] i N. i ∀ ∈

For any 1-dimensional subspace P there exists Si which do not contain P . We consider a base B of H such that each vector of B is contained in Si or P . The latter equality guarantees that the restriction of f to the associated base subset is a specialebijection of ﬁrst type, and argumentse given above imply the existence of a 1-dimensional subspace P 0 such that f([P ]) = [P 0]. Similarly, for every closed hyperplane S we choose P S and establish the i 6∈ existence of a closed hyperplane S0 such that f([S]) = [S0].

Therefore, f gives a closed collineation of ΠH . This collineation induces f.

Case (B). Since f is a special bijection of ﬁrst type, f is induced by a ◦ |B ◦ closed collineation. 68 Pankov

Acknowledgment

The result of this paper was presented on 32. Arbeitstagung uber¨ Geometrie und Algebra (9 – 11 February, 2006, Hamburg). The author thanks the organizers (A. Blunck, H. Kiechle, A. Kreuzer, H.-J. Samaga, H. Wefelscheid) for the invita- tion and ﬁnancial support.

References

[1] G. W. Mackey, Isomorphisms of normed linear spaces, Ann. of Math. 43 (1942), 244-260.

[2] M. Pankov, Transformations of Grassmannians and automorphisms of classical groups, J. Geom. 75 (2002), 132-150.

[3] , A characterization of geometrical mappings of Grassmann spaces, Results Math. 45 (2004), 319-327.

[4] , Base subsets of Grassmannians: inﬁnite-dimensional case, Eu- ropean J. Combin. 28 (2007), 26-32.

[5] J. Tits, Buildings of spherical type and ﬁnite BN-pairs, Lect. Notes in Math. 386, Springer-Verlag, 1974.

Mark Pankov DEPARTMENT OF MATHEMATICS AND INFORMATION TECHNOLOGY,, UNIVERSITY OF WARMIA AND MAZURY,, Z˙ OLNIERSKA 14A,, 10-561 OLSZTYN,, POLAND e-mail: [email protected]