I encounter the following problem of the type of degree theory in Hilbert spaces. 


Consider the Hilbert space $L^2(\mathbb T)$ with the natural inner product. 


Taking Fourier expansion with bases $e^{i2\pi k x}$, we have a direct sum decomposition into $E^-\oplus E^0\oplus E^+$, where $E^-,E^0,E^+$ correspond to subspaces with $k<0$, $k=0$ and $k>0$ respectively. 


Next take the unit sphere in the subspace $E^-\oplus E^0$, denoted by $A$, and take the ball of radius one in $E^0\oplus E^+$ centered at $(1,0)\in E^0\oplus E^+$ and denote it by $B$.


It is clear that $A$ and $B$ intersect at one point $(0,1,0)\in E^-\oplus E^0\oplus E^+$. 


Now we take the ball $B$, fix its boundary and consider any $B'$ homotopic to $B$ relative to its boundary. The question is:


Does $B'$ always intersect $A$?


Is there a topological degree theory explaining this, or a counter-example?
The Leray-Schauder degree theory does not apply to this problem. 



Topological degree in Hilbert spaces
I am not sure that I understand you correctly, but it seems that your question is answered by the following result (you can find it as Corollary 3.5 in Y.Benyamini, J.Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society,
Providence, RI, 2000): Let $E$ be an infinite-dimensional Banach space. Then there is a Lipschitz retraction of from the unit ball of $E$ to the unit sphere of $E$. 


We use this result for $E^0\oplus E^+$. Let $R:B\to\partial B$ be such retraction. Let $T\_t:B\to B$ be defined by $T\_t(x)=(1-t)x+tRx$. The image $B'\_t=T\_t(B)$ is homotopic to $B$. The Lipschitz condition implies that for $t$ close to $1$, the image does not contain $0\oplus 1\oplus 0$, which is the only candidate for the intersection of $A$ and $B'\_t$. 


