Frames associated with an operator in spaces with an indefinite metric

In the present paper, we study frames associated with an operator ( $\mathcal{W}$-frames) in Krein spaces, and we give the definition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Definition 4.1). We prove that the definition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, Ann. Funct. Anal., 14 (2023), 10.], which depends on the adjoint of the operator in the associated Hilbert space, is a consequence of our definition. We prove that our definition is independent of the fundamental decomposition (Theorem 4.1) and that having $\mathcal{W}$-frames for the Krein space necessarily gives $\mathcal{W}$-frames for the Hilbert spaces that compose the Krein space (Theorem 4.4). We also prove that orthogonal projectors generate new operators with their respective frames (Theorem 4.2). We prove an equivalence theorem for $\mathcal{W}$-frames (Theorem 4.3), without depending on the fundamental symmetry as usually given in Hilbert spaces.