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Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution with $f$). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.

$I-T$ is injective, however not invertible.

The answer is no and an example follows by amenability of $\mathbb{R}^n$ as a locally compact group.

Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution with $f$). $T$ has norm 1. 

For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. Thus the range of $I-T$ is not closed.

If the support of $f$ contains a neighborhood of the the origin then $I-T$ is injective since constant functions are not in $L^2$. However $T$ is not invertible.

Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.

$I-T$ is injective, however not invertible.

Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution with $f$). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.

$I-T$ is injective, however not invertible.

Let $\mu$ be a compactly supported probability measure defined by a density function with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=\mu(y-x)$ (i.e. $T$ acts by convolution). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.

$I-T$ is injective, however not invertible.

Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.

$I-T$ is injective, however not invertible.