Weyl criterion for the essential spectrum

If H={0}, there is no unit vector and the essential spectrum is empty, so both sides are false. Otherwise use the PVM of [A2]. For any bounded interval J and x in ran E(J), the projection identity gives E_x(R\J)=0 and $\int v^{2}d{E}_x\le ({\sup }_{v\in J}\mathrm{\mid }v\mathrm{\mid }{)}^2\mathrm{\parallel }x{\mathrm{\parallel }}^2<\mathrm{\infty }$, hence x belongs to D(A). For any x in D(A) and epsilon>0, [A2] gives $\mathrm{\parallel }(I-P_{\epsilon })x{\mathrm{\parallel }}^2=E_x(\{\mathrm{\mid }v-\lambda \mathrm{\mid }\ge \epsilon \})\le {\epsilon }^{-2}\int \mathrm{\mid }v-\lambda {\mathrm{\mid }}^{2}d{E}_x={\epsilon }^{-2}\mathrm{\parallel }(A-\lambda )x{\mathrm{\parallel }}^2$. The closed complement includes both interval endpoints.

A finite-dimensional range of an orthogonal projection P has a finite orthonormal basis e_1,...,e_r: from a finite linear basis, successively subtract its projections onto the previously obtained vectors and normalize the nonzero residuals (nonzero follows from linear independence). Then $Px={\sum }_{j=1}^r⟨x,e_j⟩e_j$, since its difference from that sum lies in the range and is orthogonal to its basis. If x_n is weakly null, [A3] gives each coefficient tending to zero and $\mathrm{\parallel }P{x}_n{\mathrm{\parallel }}^2={\sum }_{j=1}^r\mathrm{\mid }⟨x_n,e_j⟩{\mathrm{\mid }}^2\to 0$. For rank zero this is the empty sum and Px=0.

Every orthonormal sequence is weakly null: Bessel gives ${\sum }_{n\ge 0}\mathrm{\mid }⟨y,x_n⟩{\mathrm{\mid }}^2\le \mathrm{\parallel }y{\mathrm{\parallel }}^2$ for each y. If infinitely many coefficients had modulus at least epsilon>0, finite partial sums of arbitrarily many such terms would exceed this bound. Thus their moduli tend to zero; conjugate symmetry gives $⟨x_n,y⟩\to 0$, and Riesz gives convergence against every bounded linear functional.

Suppose a singular Weyl sequence exists. If any P_epsilon had finite rank, step 1.2 would give P_epsilon x_n to zero, while step 1.1 and the residual hypothesis would give (I-P_epsilon)x_n to zero. The triangle inequality would contradict norm x_n=1. Thus every P_epsilon has infinite rank, and [A1] proves lambda belongs to the essential spectrum. This proves the implication for all real lambda, including exclusion of real resolvent points, rather than merely excluding the discrete spectrum.

Suppose lambda is in the essential spectrum. For n>=0 write V_n=ran E((lambda-1/(n+1),lambda+1/(n+1))), infinite dimensional by [A1]. Given a finite list of n previously chosen orthonormal vectors x_0,...,x_(n-1), there is a nonzero vector in V_n orthogonal to them: choose n+1 linearly independent vectors in V_n and solve the n homogeneous linear equations for their pairings with the preceding vectors; a nonzero coefficient solution exists by finite-dimensional elimination, and independence makes its vector nonzero. Normalize it. For n=0 choose any nonzero vector in V_0 and normalize; there are no orthogonality equations. On the set of finite lists meeting these conditions, the relation of adjoining such a vector is entire. Apply DC from [A4] with the empty list as initial point; the compatible lists define x_n for every n>=0. The sequence is orthonormal and lies in D(A) by step 1.1. Its scalar measure is carried by the stated interval, so $\mathrm{\parallel }(A-\lambda )x_n{\mathrm{\parallel }}^2=\int \mathrm{\mid }v-\lambda {\mathrm{\mid }}^{2}d{E}_{x_n}\le (n+1{)}^{-2}\mathrm{\parallel }{x}_n{\mathrm{\parallel }}^2=(n+1{)}^{-2}$. Step 1.3 gives weak nullity. This constructs the required sequence and proves the converse.

The two implications are steps 2.1 and 2.2. Finite-dimensional H (including dimension one) has only finite-rank interval projections, so neither side holds there. Infinite multiplicity at an isolated point and spectral accumulation points are both covered by the same infinite-rank construction. Lambda=0 is allowed, since only the positive radii n+1 and epsilon are inverted. The empty initial list and the index-zero vector are included in step 2.2; the Choice use is [A4] and the spectral/Riesz interfaces, with no separability assumption.