What is the definition of dimension, you are working with?

If as ``dimension" you consider the supremum of cardinalities of  linearly independent  subsets, then the answer is ``yes". The boundedness of operator is not significant.

dim(H)=|J|, where B=\{u_j \}_{j\in J} is an orthonormal set such that \overline{span(B)}=H.

It seems to me that with this definition the answer is also positive. Let xi be elements of orthonormal basis of K. Then, for every non-zero x in H there is an i such that is non-zero. This is the same as  is non-zero, which means that T*xi form a complete system  in H. The cardinality of a complete system must be greater than or equal to the cardinality of orthonormal basis in H.

I think orthogonal its is a red herring. By the Axiom of Choice, H has an (algebraic) basis that is transformed by T to an independent subset of K, which can be extended to a basis of K.

Thanks Vladimir, you are right, the answer is positive