We propose a new class of Luenberger-like observers for LTI plants which are robust with respect to measurements affected by high-frequency noise. These observers have the classical Luenberger structure but the output injection term is corrected by means of an adaptive dead-zone. The dead-zone "cuts" part of the noise using the dead-band. The dead-zone levels are dynamically adapted according to noise, establishing a trade-off between speed of convergence and sensitivity to noise. We show that both the observer gain and the adaptation parameters can be obtained solving a Linear Matrix Inequality (LMI), whose feasibility only requires detectability of the plant. The parameters obtained through this optimization procedure ensure Uniform Global Asymptotic Stability of the origin in the error coordinates. The effectiveness of the approach is shown by means of a numerical example.