Complement of a subspace which is a cartesian product - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:26:43Z http://mathoverflow.net/feeds/question/81652 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product Complement of a subspace which is a cartesian product jjcale 2011-11-22T20:24:54Z 2011-11-23T20:38:37Z <p>Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ . Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H = U \oplus (V\times W)$ ? </p> <p>See also <a href="http://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disconnect-the-spectrum" rel="nofollow">http://mathoverflow.net/questions/24672/perturbations-of-an-operator-that-disconnect-the-spectrum</a> .</p> http://mathoverflow.net/questions/81652/complement-of-a-subspace-which-is-a-cartesian-product/81754#81754 Answer by Mikael de la Salle for Complement of a subspace which is a cartesian product Mikael de la Salle 2011-11-23T20:38:37Z 2011-11-23T20:38:37Z <p>Let me expand my comment: the answer is yes, and in general $V$ and $W$ can be constructed as follows: let $p_1,p_2:U \to H$ denote the restrictions to $U$ of the two coordinate projections. Then $V$ is the image of the spectral projection $1_{[0,1/2]}(p_1p_1^*)$ and $V$ is the image of the spectral projection $1_{[0,1/2[}(p_2 p_2^*)$ (note that one interval is open at $1/2$, whereas the other is close). One can replace the two $1/2$'s by $\delta$ and $1-\delta$, but the choice of $\delta = 1/2$ gives the best constants.</p> <p>This construction is perhaps clearer in the particular case when $U$ is a graph (i.e. <code>$U \cap (\{0\} \times H) = \{0\}$</code>, or equivalently <code>$U = \{ (x,Tx),x \in D\}$</code> for a closed operator $T$ of domain $D \subset H$). One can then easily reduce to the case when $T$ is densely defined, and (using the polar decomposition), $T$ is self-adjoint. By the spectral theorem, we can assume that there is a measure space such that $H = L^2(X,\mu)$ and $T$ is the multiplication operator by some function $f:X \to \mathbb R^+$. Then $V$ (resp. $W$) is the space of functions in $L^2(X,\mu)$ that are zero outside of $I = f^{-1}([1,\infty[)$ (resp. $J = f^{-1}([0,1[) = X \setminus I$). For $(a,b) \in H \times H$ the corresponding decomposition in $U \oplus V \times W$ is then <code>$(a 1_J + b/f 1_I , a f 1_J + b 1_I) + ((a-b/f)1_I,(b- a f)1_J)$</code>.</p>