A remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e(n)) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-Kothe spaces, then there exists a continuous unbounded quasidiagonal operator between them. Using this result, we study the corresponding Kothe matrices when every continuous linear operator between l-Kothe spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-Kothe spaces, under a splitting condition, causes the existence of a common basic subspace.


Uyanik, Elif; Yurdakul, Murat Hayrettin (2019-06-01)
For a scalar sequence (theta(n))(n is an element of N), let C be the matrix defined by c(n)(k) = theta(n-k+1) if n >= k, c(n)(k) = 0 if n < k. The map between Kothe spaces lambda(A) and lambda(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear Kothe space lambda(A) to nuclear G(1) - space lambda(B) to be linear and continuous. Its transpose is also considered.
Some finite-dimensional backward shift-invariant subspaces in the ball and a related factorization problem
Alpay, D; Kaptanoglu, HT (2000-12-15)
Beurling's theorem characterizes subspaces of the Hardy space invariant under the forward-shift operator in terms of inner functions. In this Note we consider the case where the ball replaces the open unit desk and the reproducing kernel Hilbert space with reproducing kernel 1/(1-Sigma (N)(1) a(j)w(j)*) replaces the Hardy space. We give explicit formulas which generalize Blaschke products in the case of spaces of finite codimension. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier...
Factorization of unbounded operators on Kothe spaces
Terzioglou, T; Yurdakul, Murat Hayrettin; Zuhariuta, V (2004-01-01)
The main result is that the existence of an unbounded continuous linear operator T between Kothe spaces lambda(A) and lambda(C) which factors through a third Kothe space A(B) causes the existence of an unbounded continuous quasidiagonal operator from lambda(A) into lambda(C) factoring through lambda(B) as a product of two continuous quasidiagonal operators. This fact is a factorized analogue of the Dragilev theorem [3, 6, 7, 2] about the quasidiagonal characterization of the relation (lambda(A), lambda(B)) ...
Bounded operators and complemented subspaces of Cartesian products
DJAKOV, PLAMEN; TERZİOĞLU, AHMET TOSUN; Yurdakul, Murat Hayrettin; Zahariuta, V. (2011-02-01)
We study the structure of complemented subspaces in Cartesian products X x Y of Kothe spaces X and Y under the assumption that every linear continuous operator from X to Y is bounded. In particular, it is proved that each non-Montel complemented subspace with absolute basis E subset of X x Y is isomorphic to a space of the form E(1) x E(2), where E(1) is a complemented subspace of X and E(2) is a complemented subspace of Y. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
A note on the Gauss maps of Cayley-free embeddings into spin(7)-manifolds
Ünal, İbrahim (Elsevier BV, 2018-12-01)
We show that a closed, orientable 4-manifold M admits a Cayley-free embedding into flat Spin(7)-manifold R-8 if and only if both the Euler characteristic chi(M) and the signature tau(M) of M vanish.
Citation Formats
E. Uyanik and M. H. Yurdakul, “A remark on a paper of P. B. Djakov and M. S. Ramanujan,” TURKISH JOURNAL OF MATHEMATICS, pp. 2494–2498, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30037.