We consider complex (or real) Hilbert spaces \(\mathcal {V}_1\) , \(\mathcal {V}_2\) , \(\mathcal {U}_1\) , and \(\mathcal {U}_2\) , along with bounded linear operators \(M_1: \mathcal {V}_1 \rightarrow \mathcal {U}_1\) and \(M_2: \mathcal {V}_2 \rightarrow \mathcal {U}_2\) . The direct sum space \(\mathcal {U}_1 \oplus \mathcal {U}_2\) consists of elements \(u_1 \oplus u_2\) , where \(u_1 \in \mathcal {U}_1\) and \(u_2 \in \mathcal {U}_2\) , with componentwise algebraic operations. Equipped with the inner product \( \langle u_1 \oplus u_2, v_1 \oplus v_2 \rangle := \langle u_1, v_1 \rangle _{\mathcal {U}_1} + \langle u_2, v_2 \rangle _{\mathcal {U}_2}, \) this space forms a Hilbert space. The direct sum operator \(M_1 \oplus M_2\) maps \(\mathcal {V}_1 \oplus \mathcal {V}_2\) into \(\mathcal {U}_1 \oplus \mathcal {U}_2\) via \( (M_1 \oplus M_2)(v_1 \oplus v_2) := M_1 v_1 \oplus M_2 v_2. \) In this paper, we focus on the theory of frames for operators in a general, non-classical context where domains and codomains may differ. We investigate frames for operators whose codomains are direct sums of Hilbert spaces, with special emphasis on operators of the form \(M_1 \oplus M_2\) . We analyze their duality, study their minimality properties, and extend the notion of orthonormal bases for operators to this broader setting.
