In this paper, We introduce a new class of multivariable operators known as joint m-quasihyponormal tuple of operators. It is a naturel extension of joint normal and joint hyponormal tuples of operators. An m-tuple of operators \(\mathbf{S}=(S_1, \ldots ,S_m)\in {{\mathcal {B}}}({{\mathcal {H}}})^m\) is said to be joint m-quasihyponormal tuple if \(\mathbf{S}\) satisfying

$$\begin{aligned} \displaystyle \sum _{1\le l,\;k\;\le m}\big \langle S_k^*\big [S_k^*,\;\; S_l\big ]S_lu_k\;|\;u_l\big \rangle \ge 0, \end{aligned}$$

for each finite collections \((u_l)_{1\le l\le m}\in {{\mathcal {H}}}.\) Some properties of this class of multivariable operators are studied.

中文翻译：

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Hilbert空间上的联合m-拟伪正规算子

在本文中，我们引入一个新的类被称为多变量联合运营的米商的-quasihyponormal元组。它是操作员的关节正态和关节反正态元组的自然扩展。在{{\ mathcal {B}}}（{{\ mathcal {H}}} ^ m \）中说一个m元运算符\（\ mathbf {S} =（S_1，\ ldots，S_m）\是关节米-quasihyponormal元组如果\（\ mathbf {S} \）满足

$$ \ begin {aligned} \ displaystyle \ sum _ {1 \ le l，\; k \; \ le m} \ big \ langle S_k ^ * \ big [S_k ^ *，\; \; S_l \ big] S_lu_k \; | \; u_l \ big \ rangle \ ge 0，\ end {aligned} $$

对于每个有限集合\（（u_l）_ {1 \ le l \ le m} \在{{\ mathcal {H}}}中。\）研究了这类多变量算子的一些性质。