Achievements on Matrix Hilbert spaces and reproducing kernel matrix Hilbert spaces

Abstract ‎Hilbert space is a very powerfull mathematical tool that has proven to be incredibily useful in a wide range of applications‎. ‎Matrix Hilebrt space is a new frame work that has been presented recently to perform a matrix inner product space when data observation is proposed as matrices and it is needed to analyse large dataset with complex and multi-dimensional structure‎. ‎In this paper‎, ‎a new norm associated to sequences in matrix Hilbert spaces is defined and the relation between this new norm and the norm obtained by matrix inner product is investigated‎. ‎Also‎,‎it is discussed that although‎, ‎some results and concepts in Hilbert spaces can be extended to the matrix Hilbert spaces‎, ‎but counterexamples show that the results like Pythagorean theorem‎, ‎Parallelogram law‎, ‎Jordan-Von Neumann theorem are not hold in matrix Hilbert spaces‎. ‎At last the reproducing kernel Hilbert space (RKHS) is extended to reproducing kernel matrix Hilbert space (RKMHS) and by considering some appropriate conditions on a matrix Hilbert space‎, ‎an explicit structure of reproducing kernel is obtained and the RKMHS‎, ‎H(K)‎, ‎which is obtained by reproducing kernel $K=K_{1}+K_{2}$ is characterized‎.