# Question about Fredholm's alternative theorem and norm of compact, symmetric operator

Let $$H$$ be a separable Hilbert space and K $$∈ K(H)$$ be a compact, symmetric operator.

Consider the operator $$T_μ := μI −K$$ for a given $$μ$$ ∈ ℝ∖{0}.

1. If $$|μ| > ∥K∥_{B(H)}$$ , is it true that $$T_μ$$ is a bijection?

2. Is this still true if $$|μ| = ∥K∥_{B(H)}$$ ?

For the first question i would say yes because $$∥K∥_{B(H)}$$ of a compact, symmetric operator is equal to : $$max \{|λ| : λ\ is \ eigenvalue \ of \ K\}.$$

So $$|μ|$$ > $$max$$ {$$|λ|$$} then μ cannot be an eigenvalue of $$K$$. For the Fredholm's alternative theorem this imply that $$T_μ$$ is injective and surjective.

But if $$|μ| = ∥K∥_{B(H)}$$? We have $$|μ|$$ = $$max$$ {|λ|} but i think is not true that $$μ$$ is always an eingenvalue of $$K$$ (for example $$μ$$ can be equal to -2 and the eigenvalue of $$K$$ always >0), so in principle $$T_μ$$ could be bijective. I'm not really sure about the second question. Can i have some hints?