Soft g-frames in soft Hilbert spaces

TL;DR

This paper introduces the concept of soft g-frames in soft Hilbert spaces, extending traditional frame theory to soft sets, and explores their properties, duals, and decomposition theorems.

Contribution

It extends the theory of g-frames to soft Hilbert spaces, defining soft g-frames and analyzing their properties and duals.

Findings

Soft g-frames are associated with a soft, finite, self-adjoint, reversible, and invertible g-frame operator.

Every element in soft Hilbert space admits a g-frame decomposition.

The duals of soft g-frames are characterized and constructed.

Abstract

In this paper, we define the concept of soft $g$-frame in soft Hilbert spaces by extending the concept of soft from frame to $g$-frame. We then show some properties of the soft $g$-frames in soft Hilbert spaces. Among other results, we show that the $g$-frame operator is associated with a soft, finite, self-adjoining, reversible, and finite inverse $g$-frame and get the dual g-frames. In addition, we prove that every element in Hilbert's soft space satisfies the theorem of $g$-frame decomposition.