Banach spaces

An operator acting on a Banach space is called an isometry if it preserves the norm of the space. An interesting problem is to determine the form or structure of linear isometries on Banach spaces. This can be done in some instances.
This dissertation presents several theorems that provide necessary and sufficient conditions for some linear operators acting on finite and infinite dimensional sequence spaces of complex numbers to be isometries. In all cases, the linear isometries have the form of a permutation of the elements of the sequences in the given space, with each component of each sequence multiplied by a complex number of absolute value 1.

This thesis discusses propagators and uses them to
construct a solution to the following initial value problem
under some special conditions: {du(t)/dt = A(t)u(t) u(s) = φ where A(t) is a linear operator in a Banach space X for
every t ε R, φ ε X, and s ε R.

Spectral theory, mathematical system theory, evolution equations, differential and difference equations [electronic resource] : 21st International Workshop on Operator Theory and Applications, Berlin, July 2010.It is known that lattice homomorphisms and G-solvable positive operators on Banach lattices have cyclic peripheral spectrum (See [17]). In my thesis I prove that positive contractions whose spectral radius is 1 on Banach lattices with increasing norm have cyclic peripheral point spectrum. I also prove that if the Banach lattice is a K B space satisfying the growth conditon and º is an eigenvalue of a positive contraction T such that [º] = 1, then 1 is also an eigenvalue of T as well as an eigenvalue of T¨, the dual of T. I also investigate the conditions on contraction operators on Hilbert lattices and AL-spaces which guanantee that 1 is an eigenvalue. As we know from [17], if T : E-E is a positive ideal irreducible operator on E such the r (T) = 1 is a pole of the resolvent R(º, T), then r (T) is simple pole with dimN (T -r(T)I) and ºper(T) is cyclic. Also all points of ºper(T) are simple poles of the resolvent R(º,T). SInce band irreducibility and º-order continuity do not imply ideal irreducibility [2], we prove the analogous results for band irreducible, º-order continuous operators.

Identifying and classifying the complemented subspaces of L p , p > 2, has provided much insight into the geometric structure of Lp . In 1981, Bourgain, Rosenthal, and Schechtman proved the existence of uncountably many isomorphically distinct complemented subspaces of L p , p > 2. In 1999, Dale Alspach introduced a systematic method of studying the complemented subspaces of Lp , p > 2. In this thesis, the theory of Lp spaces is developed with a concentration on techniques used to study the complemented subspaces. We define the Alspach norm and show that the possible complemented subspaces of Lp , p > 2, generated by two compatible partitions and weights are £2, £p, £2 EB £p, and(2.:EfJ £2)ep ' We have not discovered any previously unknown complemented subspaces of Lp , but this method has reduced the study and classification of these subspaces to a study of partitions of N.