The primary purpose of this paper is to discuss the g-frame operator distance problems. Specifically, let \(A \in M_{d}({\mathbb {C}})^{+}\) be a positive semidefinite \(d\times d\) complex matrix, \(a=(a_{i})_{i=1}^{n}\in ({\mathbb {R}}^{n}_{\ge 0})^{\downarrow }\) be a n-tuple of positive numbers and \({\mathbb {T}}_{d}(a)\) be a set of g-frames \(G=\{\Lambda _{i}\}_{i=1}^{n}\) such that \(\Vert T_{\Lambda _{i}}\Vert _{F}^{2}=a_{i}\), where \(T_{\Lambda _{i}}\) is a matrix of linear operator \(\Lambda _{i}\) under a standard orthogonal basis \(\{e_{k}\}_{k=1}^{n}\), for all \(i\in I_{n}\). First of all, the existence of \({\mathbb {T}}_{d}(a)\) is discussed. Then for a given matrix A, the local minimizers \(G_{0} \in {\mathbb {T}}_{d}(a)\) such that the g-frame operator distance function \(\Theta (L , A , a)=L(A-S_{G_{0}})\) gets the minimum value are studied, where \(S_{G_{0}}\) is the g-frame operator of \(G_{0}\). And some properties are obtained when \(G_{0} \in {\mathbb {T}}_{d}(a)\) is a local minimizer of \(\Theta (L , A , a)\). Furthermore, we show that a local minimizer is a global minimizer in some cases. Finally, a new definition which is called admissible is introduced. And some properties about the relationship of the sequence \(G = \{\Lambda _{i}\}_{i=1}^{n}\in {\mathbb {T}}_{d}(a)\) and an admissible pair are obtained.

本文的主要目的是讨论g帧算子距离问题。具体来说，令\（A M_ {d}（{\ mathbb {C}}）^ {+} \）是一个正半定\（d \ times d \）复矩阵\（a =（a_ {i }）_ {i = 1} ^ {n} \ in（{\ mathbb {R}} ^ {n} _ {\ ge 0}）^ {\ downarrow} \）是正数的n元组，并且\ （{\ mathbb {T}} _ {d}（a）\）是一组g帧\（G = \ {\ Lambda _ {i} \} _ {i = 1} ^ {n} \）这样\（\ Vert T _ {\ Lambda _ {i}} \ Vert _ {F} ^ {2} = a_ {i} \），其中\（T _ {\ Lambda _ {i}} \）是矩阵线性算子的\（\ LAMBDA _ {I} \）下一个标准正交基\（\ {E_ {K} \} _ {K = 1} ^ {N} \），对于所有的\（i \在I_ {n} \）中。首先，讨论\（{\ mathbb {T}} _ {d}（a）\）的存在。然后，对于给定的矩阵A，局部极小值\（G_ {0} \ in {\ mathbb {T}} _ {d}（a）\）使得g帧算子距离函数\（\ Theta（L， A，a）= L（A-S_ {G_ {0}}）\）获得最小值，其中\（S_ {G_ {0}} \）是\（G_ {0 } \）。当\（G_ {0} \ in {\ mathbb {T}} _ {d}（a）\）是\（\ Theta（L，A，a）\）的局部极小值时，可以获得一些属性。此外，我们证明了局部最小化器在某些情况下是全局最小化器。最后，引入了一个新的定义，称为可允许的。以及\（G = \ {\ Lambda _ {i} \} _ {i = 1} ^ {n} \中{\ mathbb {T}} _ {d}（a）\ ），并获得一个允许的对。