We develop the notion of the (X1,X2)-summing power-norm based on a~Banach space E, where X1 and X2 are symmetric sequence spaces. We study the particular case when X1 and X2 are Orlicz spaces ℓΦ and ℓΨ respectively and analyze under which conditions the (Φ,Ψ)-summing power-norm becomes a~multinorm. In the case when E is also a~symmetric sequence space L, we compute the precise value of ∥(δ1,⋯,δn)∥n(X1,X2), where (δk) stands for the canonical basis of L, extending known results for the (p,q)-summing power-norm based on the space ℓr which corresponds to X1=ℓp, X2=ℓq, and E=ℓr.
We develop the notion of the \((X_1,X_2)\)-summing power-norm based on a~Banach space \(E\), where \(X_1\) and \(X_2\) are symmetric sequence spaces. We study the particular case when \(X_1\) and \(X_2\) are Orlicz spaces \(\ell_\Phi\) and \(\ell_\Psi\) respectively and analyze under which conditions the \((\Phi, \Psi)\)-summing power-norm becomes a~multinorm. In the case when \(E\) is also a~symmetric sequence space \(L\), we compute the precise value of \(\|(\delta_1,\cdots,\delta_n)\|_n^{(X_1,X_2)}\) where \((\delta_k)\) stands for the canonical basis of \(L\), extending known results for the \((p,q)\)-summing power-norm based on the space \(\ell_r\) which corresponds to \(X_1=\ell_p\), \(X_2=\ell_q\), and \(E=\ell_r\).
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