MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where $(e_j)$ is an orthonormal basis of $H$ and $(x_j)$ is a family of vectors in $X$ with $\|x_j\| = 1$ and $(a_j) \in \ell_p(\mathbb{N})$.

Is there a special name for such operators? For a while I thought that these were just the absolutely $2$-summing operators between $H$ and $X$, but this seems to be wrong.

To give some background, if we have such an operator with $p=2$ and a bounded bilinear form $L$ on $X$, then the bilinear form $M$ on $H$ defined by $$M(v, w) = L(Av, Aw)$$ is trace-class, which I am interested in.

share|cite|improve this question
up vote 1 down vote accepted

See page 228ff of

Albrecht Pietsch: Operator ideals, Elsevier 1980. (pdf here)

Maybe, your operators are the $(\infty, p, \infty)$-summing operators there.

share|cite|improve this answer
Thank You for the reference. It turned out, such an operator is (2,1,2) nuclear. – Matthias Ludewig May 16 '13 at 8:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.