Let \(T\) be a bounded linear operator in a Hilbert space \(H\). A vector \(x\) in \(H\) is called hypercyclic for \(T\) if its orbit \((T^n x : n > 0)\) is dence in \(H\). The main results of the authors reads as follows: If \(T\) satisfy the hypercyclicty criterion of C. Kitai and its essential spectrum intersects the closed unit disc, then there exists an infinite-dimensional closed subspace of \(H\) consisting (expect for zero) of hypercyclic vectors of \(T\). The converse is true for operators with hypercyclic vectors not necessarily satisfying the hypercycling criterion. The authors obtain different characterizations for operators to have an infinite-dimensional closed subspace of hypercyclic vectors. The results apply to bilateral and backward weighted shifts and perturbations of the identity by backward shifts in separable Hilbert spaces, to composition operators in the Hardy space, multiplication operators in Hilbertspaces of holomorphic functions with bounded point evaluations and to the differentiation operator and translation operators in certain Hilbert spaces of entire functions. Further the authors obtain a spectral characterization of the norm-closure of the class of that operators which have an infinite-dimensional closed subspace of hypercyclic vectors. A relevant tool for the proofs is the essential minimum modulus of an operator, the author give several interesting remarks on this concept and on the essential spectrum of operators.