BOURGIN-YANG-TYPE THEOREM FOR a-COMPACT PERTURBATIONS OF CLOSED OPERATORS . PART I . THE CASE OF INDEX THEORIES WITH DIMENSION PROPERTY

1.1. Goal. Among several different, but equivalent, formulations of the famous BorsukUlam theorem, the following one is of our interest: if f : Sn → Rn is a continuous odd map, then there exists an x ∈ Sn such that f (x) = f (−x) = 0 (see [17] for other formulations, generalizations, and applications, and [11, 13] for a connection with the corresponding Brouwer degree results). Under the “stronger” assumption that f : Sn →Rm, where m < n, one can expect that there are bigger coincidence sets. The results which measure the size of the set A := {x ∈ Sn | f (−x)= f (x)} in topological terms, like dimension, (co)homology, genus (or other index theory), are usually called “Bourgin-Yang theorems.” The simplest result in this direction (cf. [5, 19]) can be formulated as follows: (i) dimA( f ) ≥ n−m (covering or cohomological dimension) and (ii) g(A( f ))≥ n−m+ 1, where g(·) stands for the genus with respect to the antipodal action (see Example 2.4). We refer to [17] for extensions of this result to more complicated (finite-dimensional) G-spaces, where G is a compact Lie group, as well as to index theories different from genus. Holm and Spanier were the first to extend the Bourgin-Yang theorem to infinite dimensions (see [10], where the solution set to the equation a(x)= f (x) was studied in the case a is a proper C∞-smooth Fredholm operator and f is a compact map; both equivariant with respect to a free involution). It should be pointed out that the assumptions on a required in [10] allow a clear finite-dimensional reduction (the kernels and images in question are complementable). At the same time, the methods developed in [10] cannot be

1. Introduction 1.1.Goal.Among several different, but equivalent, formulations of the famous Borsuk-Ulam theorem, the following one is of our interest: if f : S n → R n is a continuous odd map, then there exists an x ∈ S n such that f (x) = f (−x) = 0 (see [17] for other formulations, generalizations, and applications, and [11,13] for a connection with the corresponding Brouwer degree results).
Under the "stronger" assumption that f : S n → R m , where m < n, one can expect that there are bigger coincidence sets.The results which measure the size of the set A := {x ∈ S n | f (−x) = f (x)} in topological terms, like dimension, (co)homology, genus (or other index theory), are usually called "Bourgin-Yang theorems."The simplest result in this direction (cf.[5,19]) can be formulated as follows: (i) dimA( f ) ≥ n − m (covering or cohomological dimension) and (ii) g(A( f )) ≥ n − m + 1, where g(•) stands for the genus with respect to the antipodal action (see Example 2.4).We refer to [17] for extensions of this result to more complicated (finite-dimensional) G-spaces, where G is a compact Lie group, as well as to index theories different from genus.
Holm and Spanier were the first to extend the Bourgin-Yang theorem to infinite dimensions (see [10], where the solution set to the equation a(x) = f (x) was studied in the case a is a proper C ∞ -smooth Fredholm operator and f is a compact map; both equivariant with respect to a free involution).It should be pointed out that the assumptions on a required in [10] allow a clear finite-dimensional reduction (the kernels and images in question are complementable).At the same time, the methods developed in [10] cannot be 2 Bourgin-Yang-type theorem applied to treat the case when F is not Fredholm.The first step in this direction was done in recent papers [8,9], where the author studied the situation when a is a continuous (resp., linear closed) linear operator without any restrictions with respect to dimker(a) (in fact, in these papers only, the "dimension part" of the Bourgin-Yang theorem was proved in the presence of the antipodal symmetry).The main new ingredient in [8,9] allowing the author to go around the "complementability problem" is the application of the Michael selection theorem respecting the antipodal symmetry to the multivalued map a −1 .Observe, however, that the corresponding "equivariant selection theorem" was proved in [7] for free actions of a finite group-by no means to be extended to nonfree actions of compact Lie groups.
The main goal of our paper is to extend the results from [8][9][10] in several directions: (i) a is an arbitrary closed linear map (in general, unbounded, and having an infinite-dimensional kernel) equivariant with respect to arbitrary compact Lie group representations; (ii) f is a so-called a-compact G-equivariant map (see Definition 4.1); (iii) the coincidence set is estimated in terms of an arbitrary index theory with the so-called "dimension property" (cf.[4,17], [14,Chapter 5]).To this end, based on the results from [1], we establish a general equivariant version of the Michael selection theorem (without any restrictions with respect to G-actions) which, in our opinion, is interesting in its own.This result allows us to construct for a an equivariant section taking bounded sets to the bounded ones (see Lemma 3.6).Using this lemma, we reduce the coincidence problem to the fixed point problem.

Overview.
After Section 1, the paper is organized as follows.In Section 2, we briefly discuss "index theories."Section 3 is devoted to the proof of the equivariant Michael selection theorem and Lemma 3.6.After the reduction to the fixed point problem (see Section 4), we prove the main result (Theorem 4.3) in Section 5.In the last section, we give an application of the main result to integrodifferential equations.For the equivariant jargon, frequently used in this paper, we refer to [6].

Index theories
Convention and notations.Hereafter, G stands for a compact Lie group.
Without loss of generality, we will assume all Banach G-representations to be isometric.
Given a Banach G-representation E, (i) S R stands for the sphere in E of radius R centered at the origin; (ii) E G = {x ∈ E | gx = x, for all g ∈ G}-the fixed point set.Let us recall the standard construction of the join.Definition 2.1.Let X 1 ,...,X n be topological spaces and under the following equivalence relation: (x 1 ,...,x n ,t 1 ,...,t n ) ∼ (x 1 ,...,x n ,t 1 ,...,t n ) if and only if t i = t i (i = 1,...,n) and x i = x i whenever t i = t i > 0.
Sergey A. Antonyan et al. 3 It is convenient to denote a point of the join X 1 * ••• * X n in the form of a formal convex combination: Example 2.2.Obviously, J n S 0 = S n−1 , J n S 1 = S 2n−1 , and J n S 3 = S 4n−1 .Also, if we consider S 0 (resp., S 1 and S 3 ) as free Z 2 -(resp., S 1 -and SU(2)-spaces), then the action of Z 2 on J n S 0 (resp., S 1 on J n S 1 and SU(2) on J n S 3 ) corresponds to the antipodal action (resp., scalar multiplication in S 2n−1 ⊂ C n and scalar multiplication in S 4n−1 ⊂ H n , where H stands for the quaternions).
Following [4], [14,Chapter 5], [17], we give the following definition.Definition 2.3.A function "ind" that assigns to every G-space A a number ind(A) ∈ N ∪ {0} or {∞} is called an index theory if it satisfies the following properties.
(i) ind(A) = 0 if and only if A = ∅.
(ii) Subadditivity.If a G-space A is the union of two of its closed invariant subsets A 1 and A 2 , then ind(A) ≤ ind(A 1 ) + ind(A 2 ).(iii) Continuity.If A is a closed invariant subset of a G-space X, then there exists a closed invariant neighborhood ᐁ of A in X such that ind(A) = ind(ᐁ).(iv) Monotonicity.If A 1 and A 2 are two G-spaces and there exists an equivariant map ϕ :

Example 2.4 (genus). For a G-space A set g(
where k is minimal with this property (G acts on G/H i by left translations).If such k does not exist, put g(A) := ∞.Also, g(∅) = 0.
It is easy to check (see [3]) that the function g satisfies all the properties required for an index theory.

Definition 2.5 (dimension property). An index theory ind is said to satisfy the dimension property if there exists d
As an immediate consequence of the dimension property, one has (cf.[4]) that ind(A) Although, in general, the genus does not satisfy the dimension property, there are some important (from the application point of view) classes of groups for which it does (see the examples following below).
Remark 2.7.Restricting the genus to free G-spaces, one can define a "restricted index theory" satisfying the dimension property with d = 1 + dimG (cf.[3]).Recall that if G acts freely on a finite-dimensional sphere, then G is either finite, or S 1 , or S 3 , or the normalizer of S 1 in S 3 (cf.[6, Chapter 4, Theorem 6.2]).All finite groups admitting a free action on a finite-dimensional sphere are described in [18].

Equivariant selection theorem
We begin this section with recalling the Michael selection theorem.To this end, we need several definitions.Definition 3.1.(i) Let X and Y be topological spaces.It will be said that F is a multivalued map from X to Y if F associates with each point x ∈ X a nonempty subset is open in X.
The following fact is well known as the Michael selection theorem.Theorem 3.3 (see [15]).Let X be a paracompact space, Y a Banach space, and F an l.s.c.multivalued map from X to Y such that F(x) is a nonempty, closed, convex set for all x ∈ X.Then F admits a selection.
(the symbol on the right-hand side denotes the vector-valued integral with respect to the Haar measure).
We claim that ϕ is the desired G-selection of F. Indeed, since f (gx But the above integral belongs to conv(A f ) (see [16, Part 1, Theorem 3.27]).This yields that ϕ(x) ∈ F(x).
Continuity and equivariance of the map ϕ : X → Y can be easily derived from the corresponding properties of the integral presented in the following lemma Lemma 3.5 (see [1]).Assume that V is a complete (in the sense of the natural uniformity induced from Z) convex invariant subset of a locally convex topological vector space Z on which a compact group G acts linearly.Let C(G,V ) denote the set of all continuous maps f : G → V endowed with the compact-open topology.Then the vector-valued Haar integral : C(G,V ) → V is a well-defined continuous map satisfying the following properties: Also, assuming in addition that G is finite or Z is finite-dimensional, one can remove the completeness requirement on V .
Next, we will apply Theorem 3.4 to prove the existence of a special G-selection of a linear G-equivariant closed map of Banach G-representations.
Proof.Denote by a −1 a multivalued map from E 2 to E 1 "inverse" to a, that is, a −1 assigns to each y ∈ E 2 its full inverse image under a.Obviously, a −1 is a multivalued G-map with nonempty closed convex values.Moreover (cf.[2, Chapter 3], [8,9]), a −1 is l.s.c.(even Lipschitzian with the Lipschitz constant β(a)).
By Theorem 3.4, there exists a G-equivariant selection q : E 2 → E 1 of F. By construction, q is as required.
Remark 3.7.Lemma 3.6 is quite obvious in the case dimker(a) < ∞.Indeed, one has a direct sum decomposition E 1 = V ⊕ ker(a) and V is isomorphic to E 2 as a G-representation.However, in general, ker(a) is not complementable and, therefore, one can think of q as a nonlinear equivariant replacement for the corresponding G-isomorphism (the use of G-selections in this case seems to be unavoidable).

Main result: formulation and reduction to a fixed point problem
To formulate the main result of this paper (see Theorem 4.3), we need some preliminaries.
) is compact for any bounded sets A ⊂ E 2 and B ⊂ X (the empty set is compact by definition).
To give a simple criterion for the a-compactness of g, recall that the graph norm makes D(a) a Banach space, denoted by E. Clearly, the embedding j : E → E 1 is continuous.Put X := j −1 (X) and consider the map g : X → E 2 defined by g(x) = g( j(x)).Proposition 4.2.Under the above notations, g is a-compact if and only if g is compact.
As the proof of this proposition is straightforward, we omit it.Here is our main result.Theorem 4.3.Take an index theory ind satisfying the dimension property with some natural number d (cf.Definitions 2.3 and 2.5).Let E 1 , E 2 be Banach G-representations and 1 is a proper finite-dimensional subspace of ker(a), and denote by p the codi- The proof of Theorem 4.3 will be given in the next section.Here, by means of Lemma 3.6, we will reduce the study of (4.1) to a G-equivariant fixed point problem with a compact operator.
By assumption, E G 1 is finite-dimensional, hence we have a direct sum G-decomposition Put a := a| E1∩D(a) -the restriction.Since, by assumption, E G 1 ⊂ ker(a), we still have that a is a closed G-equivariant surjective map.Let q : E 2 → E 1 be the map provided by Lemma 3.6 (applied to a).
Next, define the map g : Further, take a direct sum G-decomposition ker(a) = E G 1 ⊕ U (dimU = p), consider the Banach G-representation E := E 2 ⊕ U equipped with diagonal G-action and the norm (y,u) = y + u , and define the map α : E → E 2 by α(y,u) := g(q(y) + u).Since q and g are equivariant, so is α.Let us show that α is a compact map.
Take a bounded set A ⊂ E. Without loss of generality, one can assume that . By the a-compactness of g, one concludes that the set Finally, take the unit sphere S ⊂ E and consider the equation Lemma 4.4.Let N(α) be the solution set to (4.4), and define the map γ : N(α) ⊂ S ⊂ E → S R ⊂ E 1 by γ(y,u) := R((q(y) + u)/ q(y) + u ).Then (i) γ is an equivariant homeomorphism onto its image; Statement (i) follows immediately from Lemma 3.6(i).To show statement (ii), take (y 0 ,u 0 ) ∈ S being a solution to (4.4).Obviously, z 0 := q(y 0 ) + u 0 = 0.By direct computation, On the other hand, using the linearity of a, one obtains Combining (4.5) and (4.6) yields x 0 := Rz 0 / z 0 ∈ N(a, f ).

Proof of the main result (Theorem 4.3)
Throughout this section, we keep the same notations as in the previous section (in particular, ker(a The proof of Theorem 4.3 splits into three steps.
Proof.By continuity property of ind, there exists a closed invariant neighborhood ᐁ ⊃ N(α) such that ind(ᐁ) = ind(N(α)).Since N(α) is compact, without loss of generality, one can assume that ᐁ is a uniform δ-neighborhood: Let us show, first, that there exists ε 0 > 0 such that N(α ε ) ⊂ ᐁ δ (N(α)) for all 0 < ε < ε 0 .Arguing indirectly, assume that for any n ∈ N, there exists (y n ,u n ) ∈ N(α 1/n ) such that y n ,u n − N(α) ≥ δ. (5.9) However, according to the definition of X and inequality (5.8), one has α(y n ,u n ) ∈ X and y n − α(y n ,u n ) < 1/n.Since X and the unit sphere of U are compact, without loss of generality, one can assume that y n → y * and u n → u * .Moreover, (y * ,u * ) ∈ S. By passing to the limit, one obtains α(y * ,u * ) = y * that contradicts (5.9).
Therefore, the statement of Lemma 5.1 follows from monotonicity property of ind.
Return to the proof of Theorem 4.3 in the considered case.Take ε small enough and the Schauder projection p ε satisfying (5.8).Let R k ⊂ E 2 be the invariant finite-dimensional subspace containing p ε (X).Put α ε := α ε | R k ⊕U and let N(α ε ) stand for the solution set to the equation α ε (y,u) = y.Combining the result obtained at the previous step with the monotonicity property of ind, one obtains (5.10) Step 3 (infinite-dimensional kernel).Under the assumptions of Theorem 4.3, suppose that p = ∞ and take a finite-dimensional invariant subspace V ⊂ U (cf. Proof.Arguing indirectly, assume that dimN(a, f ) is finite.Then N(a, f ) is compact and, therefore, ind(N(a, f )) is finite as well.The obtained contradiction completes the proof.
Remark 6.3.(i) In Proposition 6.2, one can take any index theory (for S 1 ) satisfying the dimension property.Also, the segment [0,2π] is taken to simplify the presentation.
(ii) In this paper, we restrict ourselves with the simplest illustrative example.In forthcoming papers, more involved applications (in particular, admitting closed operators with infinite-dimensional kernels) will be considered.
Below, we formulate and prove an equivariant version of the Michael selection theorem.Theorem 3.4.Let X be a paracompact G-space, Y a Banach G-representation, and F a multivalued l.s.c.G-map from X to Y such that for all x ∈ X, F(x) is a closed, convex set.Then F admits a G-selection.Proof.According to the Michael selection theorem (Theorem 3.3), there exists a continuous selection f : X → Y of F. Let dg be the normalized Haar measure on G. Define a new Sergey A. Antonyan et al. 5 single-valued map ϕ : X → Y by ϕ