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Is there some meaningful difference between: $f(x)=g(x)+c$ and $f(x)=g(x)+c\operatorname{Id}(x)$?

Because my book defines the eigenspace as follows: $$ E_\lambda=\{v\in V\mid (L-\lambda\cdot\operatorname{Id})(v)=0\}=\ker(L-\lambda\cdot\operatorname{Id}). $$ Would there be some subtle difference if I wrote $(L-\lambda)(v)$? Is the problem here that it now looks as if $\lambda$ is a function of $v$? But it is... it's the constant function on $v$.

I was just wondering if the notation I use is wrong.

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    The identity operator returns its parameter, so $\operatorname{Id} x = x$. It is a common but slight abuse of notation to write $\lambda$ instead of $\lambda \operatorname{Id}$ in the above context. – 2017-02-27

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$g(x) + c\operatorname{Id}(x)$ is exactly the same as $g(x) + cx$ because $\operatorname{Id}(x) = x$ for all $x$ in your domain.

However, $g+c\operatorname{Id}$ is not the same as $g+c$. Technically speaking, $g+c$ (function $+$ scalar) isn't even defined with the usual definitions. But, expanding on what copper.hat says in the comments, it is a common abuse of notation that literally every mathematician will understand (and most probably won't even notice).