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Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a projection), and let $T\in M$ be another self-adjoint operator.

The question will be formulated in the case of $P$ being a projection. Let $\mathcal H_0$ and $\mathcal H_1$ be the eigenspaces of $P$.

Question: Is there a well-studied condition on $P$ and $T$ which would imply that there exists $\epsilon$ such that whenever $U\subset \mathcal H$ is a "generalized eigenspace" of $T$ (i.e. the image of a spectral projection of $T$) then there exists $u\in U$ of norm $1$ such that $u=h_1+h_2$, $h_1\in \mathcal H_1$, $h_2\in \mathcal H_2$, and the norm of $h_2$ is at least $\epsilon$?

In particular:

Question: If a discrete group $\Gamma$ acts freely, ergodically and in a measure preserving way on a probablity measure space $X$, $M=L^\infty(X) \rtimes \Gamma$, $P$ is the characteristic function of some subset of $X$ of positive measure, and $T = \sum_{\gamma\in \Gamma} f_\gamma \gamma$, where $f_\gamma$ are real-valued positive functions on $X$ such that $\sum f_\gamma=1$; do $T$ and $P$ fulfill the property from the question?

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a projection), and let $T\in M$ be another self-adjoint operator.

The question will be formulated in the case of $P$ being a projection. Let $\mathcal H_0$ and $\mathcal H_1$ be the eigenspaces of $P$.

Question: Is there a well-studied condition on $P$ and $T$ which would imply that there exists $\epsilon$ such that whenever $U\subset \mathcal H$ is a "generalized eigenspace" of $T$ (i.e. the image of a spectral projection of $T$) then there exists $u\in U$ of norm $1$ such that $u=h_1+h_2$, $h_1\in \mathcal H_1$, $h_2\in \mathcal H_2$, and the norm of $h_2$ is at least $\epsilon$?

In particular (EDIT: the previous version had a weeker ergodicity assumption, in which case the answer is negative, by the answer of Steven Deprez below)

Question: If a discrete group $\Gamma$ acts freely and in a measure preserving way on a probablity measure space $X$, each element of $\Gamma$ acts ergodically, $M=L^\infty(X) \rtimes \Gamma$, $P$ is the characteristic function of some subset of $X$ of positive measure, and $T = \sum_{\gamma\in \Gamma} f_\gamma \gamma$, where $f_\gamma$ are real-valued positive functions on $X$ such that $\sum f_\gamma=1$; do $T$ and $P$ fulfill the property from the question?

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a projection), and let $T\in M$ be another self-adjoint operator.

The question will be formulated in the case of $P$ being a projection. Let $\mathcal H_0$ and $\mathcal H_1$ be the eigenspaces of $P$.

Question: Is there a well-studied condition on $P$ and $T$ which would imply that there exists $\epsilon$ such that whenever $U\subset \mathcal H$ is a "generalized eigenspace" of $T$ (i.e. the image of a spectral projection of $T$) then there exists $u\in U$ of norm $1$ such that $u=h_1+h_2$, $h_1\in \mathcal H_1$, $h_2\in \mathcal H_2$, and the norm of $h_2$ is at least $\epsilon$?

In particular, if a discrete group $\Gamma$ acts freely, ergodically and in a measure preserving way on a probablity measure space $X$, $M=L^\infty(X) \rtimes \Gamma$, $P$ is the characteristic function of some subset of $X$ of positive measure, and $T = \sum_{\gamma\in \Gamma} f_\gamma \gamma$, where $f_\gamma$ are real-valued positive functions on $X$ such that $\sum f_\gamma=1$; do $T$ and $P$ fulfill the property from the question?

Let $M\subset B(\mathcal H)$ be a finite von Neumann algebra of bounded operators on a Hilbert space $\mathcal H$., let $P\in M$ be a self-adjoint operator with a pure-point spectrum (for example a projection), and let $T\in M$ be another self-adjoint operator.

The question will be formulated in the case of $P$ being a projection. Let $\mathcal H_0$ and $\mathcal H_1$ be the eigenspaces of $P$.

Question: Is there a well-studied condition on $P$ and $T$ which would imply that there exists $\epsilon$ such that whenever $U\subset \mathcal H$ is a "generalized eigenspace" of $T$ (i.e. the image of a spectral projection of $T$) then there exists $u\in U$ of norm $1$ such that $u=h_1+h_2$, $h_1\in \mathcal H_1$, $h_2\in \mathcal H_2$, and the norm of $h_2$ is at least $\epsilon$?

In particular:

Question: If a discrete group $\Gamma$ acts freely, ergodically and in a measure preserving way on a probablity measure space $X$, $M=L^\infty(X) \rtimes \Gamma$, $P$ is the characteristic function of some subset of $X$ of positive measure, and $T = \sum_{\gamma\in \Gamma} f_\gamma \gamma$, where $f_\gamma$ are real-valued positive functions on $X$ such that $\sum f_\gamma=1$; do $T$ and $P$ fulfill the property from the question?