4 Added closeness condition of the Hilbert space under multiplication

Dear community,

I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable Hilbert space with discrete spectrum which is closed under (pointwise) pointwise multiplication. The operator I'm actually looking at is a symmetric Markov operator acting on $L^2(\mathcal{A},\mu)$, where $\mathcal{A}$ is some function algebra and $\mu$ the invariant measure.

Some questions I'm especially interested in are:

1. If you multiply two eigenfunctions, can it happen that the product has an infinite eigenfunction expansion? By "infinite eigenfunction expansion" I mean that it can not be expressed as a finite sum of eigenfunctions.

2. Somewhat related: If the squares of two eigenfunctions have a finite expansion, respectively, can it happen that the square of the sum of these two eigenfunctions has an infinite expansion?

3. In the above Markov setting: Is the following "projected Cauchy-Schwarz inequality" always true? $$\int \operatorname{proj}(fg \mid E)^2 \ \text{d} \mu \leq \sqrt{\int \operatorname{proj}(f^2 \mid E)^2 \ \text{d} \mu} \sqrt{\int \operatorname{proj}(g^2 \mid E)^2 \ \text{d} \mu}$$ Here, $f$ and $g$ are eigenfunctions lying in some common eigenspace, $E$ is another eigenspace and $\operatorname{proj}(f \mid E)$ denotes the projection of $f$ on $E$.

Note that the answer to questions 1 and 2 is no, if the eigenfunctions are orthogonal polynomials.

Simon

3 added 218 characters in body

Dear community,

I would be happy about any literature or comments on the behaviour of eigenfunctions of a self-adjoint operator on a separable Hilbert space with discrete spectrum under (pointwise) multiplication. The operator I'm actually looking at is a symmetric Markov operator acting on $L^2(\mathcal{A},\mu)$, where $\mathcal{A}$ is some function algebra and $\mu$ the invariant measure.

Some questions I'm especially interested in are:

1. If you multiply two eigenfunctions, can it happen that the product has an infinite eigenfunction expansion? By "infinite eigenfunction expansion" I mean that it can not be expressed as a finite sum of eigenfunctions.

2. Somewhat related: If the squares of two eigenfunctions have a finite expansion, respectively, can it happen that the square of the sum of these two eigenfunctions has an infinite expansion?

3. In the above Markov setting: Is the following "projected Cauchy-Schwarz inequality" always true? $$\int \operatorname{proj}(fg \mid E)^2 \ \text{d} \mu \leq \sqrt{\int \operatorname{proj}(f^2 \mid E)^2 \ \text{d} \mu} \sqrt{\int \operatorname{proj}(g^2 \mid E)^2 \ \text{d} \mu}$$ Here, $f$ and $g$ are eigenfunctions lying in some common eigenspace, $E$ is another eigenspace and $\operatorname{proj}(f \mid E)$ denotes the projection of $f$ on $E$.

Note that the answer to questions 1 and 2 is no, if the eigenfunctions are orthogonal polynomials.