Proper resolutions and Gorensteinness in triangulated categories

Let $\\mathcal {T}$ be a triangulated category with triangulation $\\Delta $, $\\xi \\subseteq \\Delta $ a proper class of triangles and $\\mathcal {C}$ an additive full subcategory of $\\mathcal {T}$. We provide a method for constructing a proper $\\mathcal {C}(\\xi )$-resolution (respectively, coproper $\\mathcal {C}(\\xi )$-coresolution) of one term in a triangle in $\\xi $ from those of the other two terms. By using this construction, we show the stability of the Gorenstein category $\\mathcal {GC}(\\xi )$ in triangulated categories. Some applications are given.