Prove that a bounded operator $A$ on Banach space is invertible (bijective) if $\left\|I - A\right\| < 1 $ holds?
68724
$||T-I|| < 1$ implies that $T$ is invertible.
140813
How do i prove that continuous linear transformation between normed spaces is bounded?
154226
Upper bound for norm of Hilbert space operator
119831
bounded operator $T$ is not compact then there exists an orthonormal sequence $e_n$ and $d>0$ such that $\|T(e_n)\|>d$ for all $n\in\Bbb{N}$?
8805
Bounded linear operators that commute with translation
81240
Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?
136909
Functional Analysis: The rank of an operator detemines if it's a compact operator.
36567
$A^{T}A$ positive definite then A is invertible?
125776
If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)
152887
Let $T : V \rightarrow V$ be a linear operator. If $T^n = O_V$ for some $n \ge 1$, prove that $I_V + T$ is an isomorphism.
71682
Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$
97621
Every separable Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$
69509
Showing a Linear Transformation is invertible
82355
When is the image of a linear operator (between Banach spaces) closed?
132491
How to show that $C=C[0,1]$ is a Banach space
33485
norm of integral operator in $C([0,1])$
52791
Invertibility of an operator $T$ if $\|I-T\|<1$